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In the traditional foundations of probability theory, one selects a probability space , and makes a distinction between *deterministic* mathematical objects, which do not depend on the sampled state , and *stochastic* (or *random*) mathematical objects, which do depend (but in a measurable fashion) on the sampled state . For instance, a *deterministic real number* would just be an element , whereas a *stochastic real number* (or *real random variable*) would be a measurable function , where in this post will always be endowed with the Borel -algebra. (For readers familiar with nonstandard analysis, the adjectives “deterministic” and “stochastic” will be used here in a manner analogous to the uses of the adjectives “standard” and “nonstandard” in nonstandard analysis. The analogy is particularly close when comparing with the “cheap nonstandard analysis” discussed in this previous blog post. We will also use “relative to ” as a synonym for “stochastic”.)

Actually, for our purposes we will adopt the philosophy of identifying stochastic objects that agree almost surely, so if one was to be completely precise, we should define a stochastic real number to be an *equivalence class* of measurable functions , up to almost sure equivalence. However, we shall often abuse notation and write simply as .

More generally, given any measurable space , we can talk either about deterministic elements , or about stochastic elements of , that is to say equivalence classes of measurable maps up to almost sure equivalence. We will use to denote the set of all stochastic elements of . (For readers familiar with sheaves, it may helpful for the purposes of this post to think of as the space of measurable global sections of the trivial -bundle over .) Of course every deterministic element of can also be viewed as a stochastic element given by (the equivalence class of) the constant function , thus giving an embedding of into . We do not attempt here to give an interpretation of for sets that are not equipped with a -algebra .

Remark 1In my previous post on the foundations of probability theory, I emphasised the freedom to extend the sample space to a larger sample space whenever one wished to inject additional sources of randomness. This is of course an important freedom to possess (and in the current formalism, is the analogue of the important operation of base change in algebraic geometry), but in this post we will focus on a single fixed sample space , and not consider extensions of this space, so that one only has to consider two types of mathematical objects (deterministic and stochastic), as opposed to having many more such types, one for each potential choice of sample space (with the deterministic objects corresponding to the case when the sample space collapses to a point).

Any (measurable) -ary operation on deterministic mathematical objects then extends to their stochastic counterparts by applying the operation pointwise. For instance, the addition operation on deterministic real numbers extends to an addition operation , by defining the class for to be the equivalence class of the function ; this operation is easily seen to be well-defined. More generally, any measurable -ary deterministic operation between measurable spaces extends to an stochastic operation in the obvious manner.

There is a similar story for -ary relations , although here one has to make a distinction between a deterministic reading of the relation and a stochastic one. Namely, if we are given stochastic objects for , the relation does not necessarily take values in the deterministic Boolean algebra , but only in the stochastic Boolean algebra – thus may be true with some positive probability and also false with some positive probability (with the event that being stochastically true being determined up to null events). Of course, the deterministic Boolean algebra embeds in the stochastic one, so we can talk about a relation being determinstically true or deterministically false, which (due to our identification of stochastic objects that agree almost surely) means that is almost surely true or almost surely false respectively. For instance given two stochastic objects , one can view their equality relation as having a stochastic truth value. This is distinct from the way the equality symbol is used in mathematical logic, which we will now call “equality in the deterministic sense” to reduce confusion. Thus, in the deterministic sense if and only if the stochastic truth value of is equal to , that is to say that for almost all .

Any universal identity for deterministic operations (or universal implication between identities) extends to their stochastic counterparts: for instance, addition is commutative, associative, and cancellative on the space of deterministic reals , and is therefore commutative, associative, and cancellative on stochastic reals as well. However, one has to be more careful when working with mathematical laws that are not expressible as universal identities, or implications between identities. For instance, is an integral domain: if are deterministic reals such that , then one must have or . However, if are *stochastic* reals such that (in the deterministic sense), then it is no longer necessarily the case that (in the deterministic sense) or that (in the deterministic sense); however, it is still true that “ or ” is true in the deterministic sense if one interprets the boolean operator “or” stochastically, thus “ or ” is true for almost all . Another way to properly obtain a stochastic interpretation of the integral domain property of is to rewrite it as

and then make all sets stochastic to obtain the true statement

thus we have to allow the index for which vanishing occurs to also be stochastic, rather than deterministic. (A technical note: when one proves this statement, one has to select in a measurable fashion; for instance, one can choose to equal when , and otherwise (so that in the “tie-breaking” case when and both vanish, one always selects to equal ).)

Similarly, the law of the excluded middle fails when interpreted deterministically, but remains true when interpreted stochastically: if is a stochastic statement, then it is not necessarily the case that is either deterministically true or deterministically false; however the sentence “ or not-” is still deterministically true if the boolean operator “or” is interpreted stochastically rather than deterministically.

To avoid having to keep pointing out which operations are interpreted stochastically and which ones are interpreted deterministically, we will use the following convention: if we assert that a mathematical sentence involving stochastic objects is true, then (unless otherwise specified) we mean that is deterministically true, assuming that all relations used inside are interpreted stochastically. For instance, if are stochastic reals, when we assert that “Exactly one of , , or is true”, then by default it is understood that the relations , , and the boolean operator “exactly one of” are interpreted stochastically, and the assertion is that the sentence is deterministically true.

In the above discussion, the stochastic objects being considered were elements of a deterministic space , such as the reals . However, it can often be convenient to generalise this situation by allowing the ambient space to also be stochastic. For instance, one might wish to consider a stochastic vector inside a stochastic vector space , or a stochastic edge of a stochastic graph . In order to formally describe this situation within the classical framework of measure theory, one needs to place all the ambient spaces inside a measurable space. This can certainly be done in many contexts (e.g. when considering random graphs on a deterministic set of vertices, or if one is willing to work up to equivalence and place the ambient spaces inside a suitable moduli space), but is not completely natural in other contexts. For instance, if one wishes to consider stochastic vector spaces of potentially unbounded dimension (in particular, potentially larger than any given cardinal that one might specify in advance), then the class of all possible vector spaces is so large that it becomes a proper class rather than a set (even if one works up to equivalence), making it problematic to give this class the structure of a measurable space; furthermore, even once one does so, one needs to take additional care to pin down what it would mean for a random vector lying in a random vector space to depend “measurably” on .

Of course, in any reasonable application one can avoid the set theoretic issues at least by various *ad hoc* means, for instance by restricting the dimension of all spaces involved to some fixed cardinal such as . However, the measure-theoretic issues can require some additional effort to resolve properly.

In this post I would like to describe a different way to formalise stochastic spaces, and stochastic elements of these spaces, by viewing the spaces as measure-theoretic analogue of a sheaf, but being over the probability space rather than over a topological space; stochastic objects are then *sections* of such sheaves. Actually, for minor technical reasons it is convenient to work in the slightly more general setting in which the base space is a finite measure space rather than a probability space, thus can take any value in rather than being normalised to equal . This will allow us to easily localise to subevents of without the need for normalisation, even when is a null event (though we caution that the map from deterministic objects ceases to be injective in this latter case). We will however still continue to use probabilistic terminology. despite the lack of normalisation; thus for instance, sets in will be referred to as *events*, the measure of such a set will be referred to as the *probability* (which is now permitted to exceed in some cases), and an event whose complement is a null event shall be said to hold *almost surely*. It is in fact likely that almost all of the theory below extends to base spaces which are -finite rather than finite (for instance, by damping the measure to become finite, without introducing any further null events), although we will not pursue this further generalisation here.

The approach taken in this post is “topos-theoretic” in nature (although we will not use the language of topoi explicitly here), and is well suited to a “pointless” or “point-free” approach to probability theory, in which the role of the stochastic state is suppressed as much as possible; instead, one strives to always adopt a “relative point of view”, with all objects under consideration being viewed as stochastic objects relative to the underlying base space . In this perspective, the stochastic version of a set is as follows.

Definition 1 (Stochastic set)Unless otherwise specified, we assume that we are given a fixed finite measure space (which we refer to as thebase space). Astochastic set(relative to ) is a tuple consisting of the following objects:

- A set assigned to each event ; and
- A
restriction mapfrom to to each pair of nested events . (Strictly speaking, one should indicate the dependence on in the notation for the restriction map, e.g. using instead of , but we will abuse notation by omitting the dependence.)We refer to elements of as

local stochastic elementsof the stochastic set , localised to the event , and elements of asglobal stochastic elements(or simplyelements) of the stochastic set. (In the language of sheaves, one would use “sections” instead of “elements” here, but I prefer to use the latter terminology here, for compatibility with conventional probabilistic notation, where for instance measurable maps from to are referred to as real random variables, rather than sections of the reals.)Furthermore, we impose the following axioms:

- (Category) The map from to is the identity map, and if are events in , then for all .
- (Null events trivial) If is a null event, then the set is a singleton set. (In particular, is always a singleton set; this is analogous to the convention that for any number .)
- (Countable gluing) Suppose that for each natural number , one has an event and an element such that for all . Then there exists a unique such that for all .
If is an event in , we define the

localisationof the stochastic set to to be the stochastic setrelative to . (Note that there is no need to renormalise the measure on , as we are not demanding that our base space have total measure .)

The following fact is useful for actually verifying that a given object indeed has the structure of a stochastic set:

Exercise 1Show that to verify the countable gluing axiom of a stochastic set, it suffices to do so under the additional hypothesis that the events are disjoint. (Note that this is quite different from the situation with sheaves over a topological space, in which the analogous gluing axiom is often trivial in the disjoint case but has non-trivial content in the overlapping case. This is ultimately because a -algebra is closed under all Boolean operations, whereas a topology is only closed under union and intersection.)

Let us illustrate the concept of a stochastic set with some examples.

Example 1 (Discrete case)A simple case arises when is a discrete space which is at most countable. If we assign a set to each , with a singleton if . One then sets , with the obvious restriction maps, giving rise to a stochastic set . (Thus, a local element of can be viewed as a map on that takes values in for each .) Conversely, it is not difficult to see that any stochastic set over an at most countable discrete probability space is of this form up to isomorphism. In this case, one can think of as a bundle of sets over each point (of positive probability) in the base space . One can extend this bundle interpretation of stochastic sets to reasonably nice sample spaces (such as standard Borel spaces) and similarly reasonable ; however, I would like to avoid this interpretation in the formalism below in order to be able to easily work in settings in which and are very “large” (e.g. not separable in any reasonable sense). Note that we permit some of the to be empty, thus it can be possible for to be empty whilst for some strict subevents of to be non-empty. (This is analogous to how it is possible for a sheaf to have local sections but no global sections.) As such, the space of global elements does not completely determine the stochastic set ; one sometimes needs to localise to an event in order to see the full structure of such a set. Thus it is important to distinguish between a stochastic set and its space of global elements. (As such, it is a slight abuse of the axiom of extensionality to refer to global elements of simply as “elements”, but hopefully this should not cause too much confusion.)

Example 2 (Measurable spaces as stochastic sets)Returning now to a general base space , any (deterministic) measurable space gives rise to a stochastic set , with being defined as in previous discussion as the measurable functions from to modulo almost everywhere equivalence (in particular, a singleton set when is null), with the usual restriction maps. The constraint of measurability on the maps , together with the quotienting by almost sure equivalence, means that is now more complicated than a plain Cartesian product of fibres, but this still serves as a useful first approximation to what is for the purposes of developing intuition. Indeed, the measurability constraint is so weak (as compared for instance to topological or smooth constraints in other contexts, such as sheaves of continuous or smooth sections of bundles) that the intuition of essentially independent fibres is quite an accurate one, at least if one avoids consideration of an uncountable number of objects simultaneously.

Example 3 (Extended Hilbert modules)This example is the one that motivated this post for me. Suppose that one has an extension of the base space , thus we have a measurable factor map such that the pushforward of the measure by is equal to . Then we have a conditional expectation operator , defined as the adjoint of the pullback map . As is well known, the conditional expectation operator also extends to a contraction ; by monotone convergence we may also extend to a map from measurable functions from to the extended non-negative reals , to measurable functions from to . We then define the “extended Hilbert module” to be the space of functions with finite almost everywhere. This is an extended version of the Hilbert module , which is defined similarly except that is required to lie in ; this is a Hilbert module over which is of particular importance in the Furstenberg-Zimmer structure theory of measure-preserving systems. We can then define the stochastic set by settingwith the obvious restriction maps. In the case that are standard Borel spaces, one can

disintegrateas an integral of probability measures (supported in the fibre ), in which case this stochastic set can be viewed as having fibres (though if is not discrete, there are still some measurability conditions in on the local and global elements that need to be imposed). However, I am interested in the case when are not standard Borel spaces (in fact, I will take them to be algebraic probability spaces, as defined in this previous post), in which case disintegrations are not available. However, it appears that the stochastic analysis developed in this blog post can serve as a substitute for the tool of disintegration in this context.

We make the remark that if is a stochastic set and are events that are equivalent up to null events, then one can identify with (through their common restriction to , with the restriction maps now being bijections). As such, the notion of a stochastic set does not require the full structure of a concrete probability space ; one could also have defined the notion using only the abstract -algebra consisting of modulo null events as the base space, or equivalently one could define stochastic sets over the algebraic probability spaces defined in this previous post. However, we will stick with the classical formalism of concrete probability spaces here so as to keep the notation reasonably familiar.

As a corollary of the above observation, we see that if the base space has total measure , then all stochastic sets are trivial (they are just points).

Exercise 2If is a stochastic set, show that there exists an event with the property that for any event , is non-empty if and only if is contained in modulo null events. (In particular, is unique up to null events.)Hint:consider the numbers for ranging over all events with non-empty, and form a maximising sequence for these numbers. Then use all three axioms of a stochastic set.

One can now start take many of the fundamental objects, operations, and results in set theory (and, hence, in most other categories of mathematics) and establish analogues relative to a finite measure space. Implicitly, what we will be doing in the next few paragraphs is endowing the category of stochastic sets with the structure of an elementary topos. However, to keep things reasonably concrete, we will not explicitly emphasise the topos-theoretic formalism here, although it is certainly lurking in the background.

Firstly, we define a *stochastic function* between two stochastic sets to be a collection of maps for each which form a natural transformation in the sense that for all and nested events . In the case when is discrete and at most countable (and after deleting all null points), a stochastic function is nothing more than a collection of functions for each , with the function then being a direct sum of the factor functions :

Thus (in the discrete, at most countable setting, at least) stochastic functions do not mix together information from different states in a sample space; the value of at depends only on the value of at . The situation is a bit more subtle for continuous probability spaces, due to the identification of stochastic objects that agree almost surely, nevertheness it is still good intuition to think of stochastic functions as essentially being “pointwise” or “local” in nature.

One can now form the stochastic set of functions from to , by setting for any event to be the set of local stochastic functions of the localisations of to ; this is a stochastic set if we use the obvious restriction maps. In the case when is discrete and at most countable, the fibre at a point of positive measure is simply the set of functions from to .

In a similar spirit, we say that one stochastic set is a (stochastic) subset of another , and write , if we have a stochastic inclusion map, thus for all events , with the restriction maps being compatible. We can then define the power set of a stochastic set by setting for any event to be the set of all stochastic subsets of relative to ; it is easy to see that is a stochastic set with the obvious restriction maps (one can also identify with in the obvious fashion). Again, when is discrete and at most countable, the fibre of at a point of positive measure is simply the deterministic power set .

Note that if is a stochastic function and is a stochastic subset of , then the inverse image , defined by setting for any event to be the set of those with , is a stochastic subset of . In particular, given a -ary relation , the inverse image is a stochastic subset of , which by abuse of notation we denote as

In a similar spirit, if is a stochastic subset of and is a stochastic function, we can define the image by setting to be the set of those with ; one easily verifies that this is a stochastic subset of .

Remark 2One should caution that in the definition of the subset relation , it is important that for all events , not just the global event ; in particular, just because a stochastic set has no global sections, does not mean that it is contained in the stochastic empty set .

Now we discuss Boolean operations on stochastic subsets of a given stochastic set . Given two stochastic subsets of , the stochastic intersection is defined by setting to be the set of that lie in both and :

This is easily verified to again be a stochastic subset of . More generally one may define stochastic countable intersections for any sequence of stochastic subsets of . One *could* extend this definition to uncountable families if one wished, but I would advise against it, because some of the usual laws of Boolean algebra (e.g. the de Morgan laws) may break down in this setting.

Stochastic unions are a bit more subtle. The set should not be defined to simply be the union of and , as this would not respect the gluing axiom. Instead, we define to be the set of all such that one can cover by measurable subevents such that for ; then may be verified to be a stochastic subset of . Thus for instance is the stochastic union of and . Similarly for countable unions of stochastic subsets of , although for uncountable unions are extremely problematic (they are disliked by both the measure theory and the countable gluing axiom) and will not be defined here. Finally, the stochastic difference set is defined as the set of all in such that for any subevent of of positive probability. One may verify that in the case when is discrete and at most countable, these Boolean operations correspond to the classical Boolean operations applied separately to each fibre of the relevant sets . We also leave as an exercise to the reader to verify the usual laws of Boolean arithmetic, e.g. the de Morgan laws, provided that one works with at most countable unions and intersections.

One can also consider a stochastic finite union in which the number of sets in the union is itself stochastic. More precisely, let be a stochastic set, let be a stochastic natural number, and let be a stochastic function from the stochastic set (defined by setting )) to the stochastic power set . Here we are considering to be a natural number, to allow for unions that are possibly empty, with used for the positive natural numbers. We also write for the stochastic function . Then we can define the stochastic union by setting for an event to be the set of local elements with the property that there exists a covering of by measurable subevents for , such that one has and . One can verify that is a stochastic set (with the obvious restriction maps). Again, in the model case when is discrete and at most countable, the fibre is what one would expect it to be, namely .

The Cartesian product of two stochastic sets may be defined by setting for all events , with the obvious restriction maps; this is easily seen to be another stochastic set. This lets one define the concept of a -ary operation from stochastic sets to another stochastic set , or a -ary relation . In particular, given for , the relation may be deterministically true, deterministically false, or have some other stochastic truth value.

Remark 3In the degenerate case when is null, stochastic logic becomes a bit weird: all stochastic statements are deterministically true, as are their stochastic negations, since every event in (even the empty set) now holds with full probability. Among other pathologies, the empty set now has a global element over (this is analogous to the notorious convention ), and any two deterministic objects become equal over : .

The following simple observation is crucial to subsequent discussion. If is a sequence taking values in the global elements of a stochastic space , then we may also define global elements for *stochastic* indices as well, by appealing to the countable gluing axiom to glue together restricted to the set for each deterministic natural number to form . With this definition, the map is a stochastic function from to ; indeed, this creates a one-to-one correspondence between external sequences (maps from to ) and stochastic sequences (stochastic functions from to ). Similarly with replaced by any other at most countable set. This observation will be important in allowing many deterministic arguments involving sequences will be able to be carried over to the stochastic setting.

We now specialise from the extremely broad discipline of set theory to the more focused discipline of real analysis. There are two fundamental axioms that underlie real analysis (and in particular distinguishes it from real *algebra*). The first is the Archimedean property, which we phrase in the “no infinitesimal” formulation as follows:

Proposition 2 (Archimedean property)Let be such that for all positive natural numbers . Then .

The other is the least upper bound axiom:

Proposition 3 (Least upper bound axiom)Let be a non-empty subset of which has an upper bound , thus for all . Then there exists a unique real number with the following properties:

- for all .
- For any real , there exists such that .
- .
Furthermore, does not depend on the choice of .

The Archimedean property extends easily to the stochastic setting:

Proposition 4 (Stochastic Archimedean property)Let be such that for all deterministic natural numbers . Then .

Remark 4Here, incidentally, is one place in which this stochastic formalism deviates from the nonstandard analysis formalism, as the latter certainly permits the existence of infinitesimal elements. On the other hand, we caution that stochastic real numbers are permitted to be unbounded, so that formulation of Archimedean property is not valid in the stochastic setting.

The proof is easy and is left to the reader. The least upper bound axiom also extends nicely to the stochastic setting, but the proof requires more work (in particular, our argument uses the monotone convergence theorem):

Theorem 5 (Stochastic least upper bound axiom)Let be a stochastic subset of which has a global upper bound , thus for all , and is globally non-empty in the sense that there is at least one global element . Then there exists a unique stochastic real number with the following properties:

- for all .
- For any stochastic real , there exists such that .
- .
Furthermore, does not depend on the choice of .

For future reference, we note that the same result holds with replaced by throughout, since the latter may be embedded in the former, for instance by mapping to and to . In applications, the above theorem serves as a reasonable substitute for the countable axiom of choice, which does not appear to hold in unrestricted generality relative to a measure space; in particular, it can be used to generate various extremising sequences for stochastic functionals on various stochastic function spaces.

*Proof:* Uniqueness is clear (using the Archimedean property), as well as the independence on , so we turn to existence. By using an order-preserving map from to (e.g. ) we may assume that is a subset of , and that .

We observe that is a lattice: if , then and also lie in . Indeed, may be formed by appealing to the countable gluing axiom to glue (restricted the set ) with (restricted to the set ), and similarly for . (Here we use the fact that relations such as are Borel measurable on .)

Let denote the deterministic quantity

then (by Proposition 3!) is well-defined; here we use the hypothesis that is finite. Thus we may find a sequence of elements of such that

Using the lattice property, we may assume that the are non-decreasing: whenever . If we then define (after choosing measurable representatives of each equivalence class ), then is a stochastic real with .

If , then , and so

From this and (1) we conclude that

From monotone convergence, we conclude that

and so , as required.

Now let be a stochastic real. After choosing measurable representatives of each relevant equivalence class, we see that for almost every , we can find a natural number with . If we choose to be the first such positive natural number when it exists, and (say) otherwise, then is a stochastic positive natural number and . The claim follows.

Remark 5One can abstract away the role of the measure here, leaving only the ideal of null sets. The property that the measure is finite is then replaced by the more general property that given any non-empty family of measurable sets, there is an at most countable union of sets in that family that is an upper bound modulo null sets for all elements in that faily.

Using Proposition 4 and Theorem 5, one can then revisit many of the other foundational results of deterministic real analysis, and develop stochastic analogues; we give some examples of this below the fold (focusing on the Heine-Borel theorem and a case of the spectral theorem). As an application of this formalism, we revisit some of the Furstenberg-Zimmer structural theory of measure-preserving systems, particularly that of relatively compact and relatively weakly mixing systems, and interpret them in this framework, basically as stochastic versions of compact and weakly mixing systems (though with the caveat that the shift map is allowed to act non-trivially on the underlying probability space). As this formalism is “point-free”, in that it avoids explicit use of fibres and disintegrations, it will be well suited for generalising this structure theory to settings in which the underlying probability spaces are not standard Borel, and the underlying groups are uncountable; I hope to discuss such generalisations in future blog posts.

Remark 6Roughly speaking, stochastic real analysis can be viewed as a restricted subset of classical real analysis in which all operations have to be “measurable” with respect to the base space. In particular, indiscriminate application of the axiom of choice is not permitted, and one should largely restrict oneself to performing countable unions and intersections rather than arbitrary unions or intersections. Presumably one can formalise this intuition with a suitable “countable transfer principle”, but I was not able to formulate a clean and general principle of this sort, instead verifying various assertions about stochastic objects by hand rather than by direct transfer from the deterministic setting. However, it would be desirable to have such a principle, since otherwise one is faced with the tedious task of redoing all the foundations of real analysis (or whatever other base theory of mathematics one is going to be working in) in the stochastic setting by carefully repeating all the arguments.More generally, topos theory is a good formalism for capturing precisely the informal idea of performing mathematics with certain operations, such as the axiom of choice, the law of the excluded middle, or arbitrary unions and intersections, being somehow “prohibited” or otherwise “restricted”.

Due to some requests, I’m uploading to my blog the slides for my recent talk in Segovia (for the birthday conference of Michael Cowling) on “Hilbert’s fifth problem and approximate groups“. The slides cover essentially the same range of topics in this series of lecture notes, or in this text of mine, though of course in considerably less detail, given that the slides are meant to be presented in an hour.

As laid out in the foundational work of Kolmogorov, a *classical probability space* (or probability space for short) is a triplet , where is a set, is a -algebra of subsets of , and is a countably additive probability measure on . Given such a space, one can form a number of interesting function spaces, including

- the (real) Hilbert space of square-integrable functions , modulo -almost everywhere equivalence, and with the positive definite inner product ; and
- the unital commutative Banach algebra of essentially bounded functions , modulo -almost everywhere equivalence, with defined as the essential supremum of .

There is also a trace on defined by integration: .

One can form the category of classical probability spaces, by defining a morphism between probability spaces to be a function which is measurable (thus for all ) and measure-preserving (thus for all ).

Let us now abstract the algebraic features of these spaces as follows; for want of a better name, I will refer to this abstraction as an *algebraic probability space*, and is very similar to the non-commutative probability spaces studied in this previous post, except that these spaces are now commutative (and real).

Definition 1Analgebraic probability spaceis a pair where

- is a unital commutative real algebra;
- is a homomorphism such that and for all ;
- Every element of is
boundedin the sense that . (Technically, this isn’t an algebraic property, but I need it for technical reasons.)A morphism is a homomorphism which is trace-preserving, in the sense that for all .

For want of a better name, I’ll denote the category of algebraic probability spaces as . One can view this category as the opposite category to that of (a subcategory of) the category of tracial commutative real algebras. One could emphasise this opposite nature by denoting the algebraic probability space as rather than ; another suggestive (but slightly inaccurate) notation, inspired by the language of schemes, would be rather than . However, we will not adopt these conventions here, and refer to algebraic probability spaces just by the pair .

By the previous discussion, we have a covariant functor that takes a classical probability space to its algebraic counterpart , with a morphism of classical probability spaces mapping to a morphism of the corresponding algebraic probability spaces by the formula

for . One easily verifies that this is a functor.

In this post I would like to describe a functor which partially inverts (up to natural isomorphism), that is to say a recipe for starting with an algebraic probability space and producing a classical probability space . This recipe is not new – it is basically the (commutative) Gelfand-Naimark-Segal construction (discussed in this previous post) combined with the Loomis-Sikorski theorem (discussed in this previous post). However, I wanted to put the construction in a single location for sake of reference. I also wanted to make the point that and are not complete inverses; there is a bit of information in the algebraic probability space (e.g. topological information) which is lost when passing back to the classical probability space. In some future posts, I would like to develop some ergodic theory using the algebraic foundations of probability theory rather than the classical foundations; this turns out to be convenient in the ergodic theory arising from nonstandard analysis (such as that described in this previous post), in which the groups involved are uncountable and the underlying spaces are not standard Borel spaces.

Let us describe how to construct the functor , with details postponed to below the fold.

- Starting with an algebraic probability space , form an inner product on by the formula , and also form the spectral radius .
- The inner product is clearly positive semi-definite. Quotienting out the null vectors and taking completions, we arrive at a real Hilbert space , to which the trace may be extended.
- Somewhat less obviously, the spectral radius is well-defined and gives a norm on . Taking limits of sequences in of bounded spectral radius gives us a subspace of that has the structure of a real commutative Banach algebra.
- The idempotents of the Banach algebra may be indexed by elements of an abstract -algebra .
- The Boolean algebra homomorphisms (or equivalently, the real algebra homomorphisms ) may be indexed by elements of a space .
- Let denote the -algebra on generated by the basic sets for every .
- Let be the -ideal of generated by the sets , where is a sequence with .
- One verifies that is isomorphic to . Using this isomorphism, the trace on can be used to construct a countably additive measure on . The classical probability space is then , and the abstract spaces may now be identified with their concrete counterparts , .
- Every algebraic probability space morphism generates a classical probability morphism via the formula
using a pullback operation on the abstract -algebras that can be defined by density.

Remark 1The classical probability space constructed by the functor has some additional structure; namely is a -Stone space (a Stone space with the property that the closure of any countable union of clopen sets is clopen), is the Baire -algebra (generated by the clopen sets), and the null sets are the meager sets. However, we will not use this additional structure here.

The partial inversion relationship between the functors and is given by the following assertion:

- There is a natural transformation from to the identity functor .

More informally: if one starts with an algebraic probability space and converts it back into a classical probability space , then there is a trace-preserving algebra homomorphism of to , which respects morphisms of the algebraic probability space. While this relationship is far weaker than an equivalence of categories (which would require that and are both natural isomorphisms), it is still good enough to allow many ergodic theory problems formulated using classical probability spaces to be reformulated instead as an equivalent problem in algebraic probability spaces.

Remark 2The opposite composition is a little odd: it takes an arbitrary probability space and returns a more complicated probability space , with being the space of homomorphisms . while there is “morally” an embedding of into using the evaluation map, this map does not exist in general because points in may well have zero measure. However, if one takes a “pointless” approach and focuses just on the measure algebras , , then these algebras become naturally isomorphic after quotienting out by null sets.

Remark 3An algebraic probability space captures a bit more structure than a classical probability space, because may be identified with a proper subset of that describes the “regular” functions (or random variables) of the space. For instance, starting with the unit circle (with the usual Haar measure and the usual trace ), any unital subalgebra of that is dense in will generate the same classical probability space on applying the functor , namely one will get the space of homomorphisms from to (with the measure induced from ). Thus for instance could be the continuous functions , the Wiener algebra or the full space , but the classical space will be unable to distinguish these spaces from each other. In particular, the functor loses information (roughly speaking, this functor takes an algebraic probability space and completes it to a von Neumann algebra, but then forgets exactly what algebra was initially used to create this completion). In ergodic theory, this sort of “extra structure” is traditionally encoded in topological terms, by assuming that the underlying probability space has a nice topological structure (e.g. a standard Borel space); however, with the algebraic perspective one has the freedom to have non-topological notions of extra structure, by choosing to be something other than an algebra of continuous functions on a topological space. I hope to discuss one such example of extra structure (coming from the Gowers-Host-Kra theory of uniformity seminorms) in a later blog post (this generalises the example of the Wiener algebra given previously, which is encoding “Fourier structure”).

A small example of how one could use the functors is as follows. Suppose one has a classical probability space with a measure-preserving action of an uncountable group , which is only defined (and an action) up to almost everywhere equivalence; thus for instance for any set and any , and might not be exactly equal, but only equal up to a null set. For similar reasons, an element of the invariant factor might not be exactly invariant with respect to , but instead one only has and equal up to null sets for each . One might like to “clean up” the action of to make it defined everywhere, and a genuine action everywhere, but this is not immediately achievable if is uncountable, since the union of all the null sets where something bad occurs may cease to be a null set. However, by applying the functor , each shift defines a morphism on the associated algebraic probability space (i.e. the Koopman operator), and then applying , we obtain a shift on a new classical probability space which now gives a genuine measure-preserving action of , and which is equivalent to the original action from a measure algebra standpoint. The invariant factor now consists of those sets in which are genuinely -invariant, not just up to null sets. (Basically, the classical probability space contains a Boolean algebra with the property that every measurable set is equivalent up to null sets to precisely one set in , allowing for a canonical “retraction” onto that eliminates all null set issues.)

More indirectly, the functors suggest that one should be able to develop a “pointless” form of ergodic theory, in which the underlying probability spaces are given algebraically rather than classically. I hope to give some more specific examples of this in later posts.

There are a number of ways to construct the real numbers , for instance

- as the metric completion of (thus, is defined as the set of Cauchy sequences of rationals, modulo Cauchy equivalence);
- as the space of Dedekind cuts on the rationals ;
- as the space of quasimorphisms on the integers, quotiented by bounded functions. (I believe this construction first appears in this paper of Street, who credits the idea to Schanuel, though the germ of this construction arguably goes all the way back to Eudoxus.)

There is also a fourth family of constructions that proceeds via nonstandard analysis, as a special case of what is known as the *nonstandard hull construction*. (Here I will assume some basic familiarity with nonstandard analysis and ultraproducts, as covered for instance in this previous blog post.) Given an unbounded nonstandard natural number , one can define two external additive subgroups of the nonstandard integers :

- The group of all nonstandard integers of magnitude less than or comparable to ; and
- The group of nonstandard integers of magnitude infinitesimally smaller than .

The group is a subgroup of , so we may form the quotient group . This space is isomorphic to the reals , and can in fact be used to construct the reals:

Proposition 1For any coset of , there is a unique real number with the property that . The map is then an isomorphism between the additive groups and .

*Proof:* Uniqueness is clear. For existence, observe that the set is a Dedekind cut, and its supremum can be verified to have the required properties for .

In a similar vein, we can view the unit interval in the reals as the quotient

where is the nonstandard (i.e. internal) set ; of course, is not a group, so one should interpret as the image of under the quotient map (or , if one prefers). Or to put it another way, (1) asserts that is the image of with respect to the map .

In this post I would like to record a nice measure-theoretic version of the equivalence (1), which essentially appears already in standard texts on Loeb measure (see e.g. this text of Cutland). To describe the results, we must first quickly recall the construction of *Loeb measure* on . Given an internal subset of , we may define the elementary measure of by the formula

This is a finitely additive probability measure on the Boolean algebra of internal subsets of . We can then construct the *Loeb outer measure* of any subset in complete analogy with Lebesgue outer measure by the formula

where ranges over all sequences of internal subsets of that cover . We say that a subset of is *Loeb measurable* if, for any (standard) , one can find an internal subset of which differs from by a set of Loeb outer measure at most , and in that case we define the *Loeb measure* of to be . It is a routine matter to show (e.g. using the Carathéodory extension theorem) that the space of Loeb measurable sets is a -algebra, and that is a countably additive probability measure on this space that extends the elementary measure . Thus now has the structure of a probability space .

Now, the group acts (Loeb-almost everywhere) on the probability space by the addition map, thus for and (excluding a set of Loeb measure zero where exits ). This action is clearly seen to be measure-preserving. As such, we can form the *invariant factor* , defined by restricting attention to those Loeb measurable sets with the property that is equal -almost everywhere to for each .

The claim is then that this invariant factor is equivalent (up to almost everywhere equivalence) to the unit interval with Lebesgue measure (and the trivial action of ), by the same factor map used in (1). More precisely:

Theorem 2Given a set , there exists a Lebesgue measurable set , unique up to -a.e. equivalence, such that is -a.e. equivalent to the set . Conversely, if is Lebesgue measurable, then is in , and .

More informally, we have the measure-theoretic version

of (1).

*Proof:* We first prove the converse. It is clear that is -invariant, so it suffices to show that is Loeb measurable with Loeb measure . This is easily verified when is an elementary set (a finite union of intervals). By countable subadditivity of outer measure, this implies that Loeb outer measure of is bounded by the Lebesgue outer measure of for any set ; since every Lebesgue measurable set differs from an elementary set by a set of arbitrarily small Lebesgue outer measure, the claim follows.

Now we establish the forward claim. Uniqueness is clear from the converse claim, so it suffices to show existence. Let . Let be an arbitrary standard real number, then we can find an internal set which differs from by a set of Loeb measure at most . As is -invariant, we conclude that for every , and differ by a set of Loeb measure (and hence elementary measure) at most . By the (contrapositive of the) underspill principle, there must exist a standard such that and differ by a set of elementary measure at most for all . If we then define the nonstandard function by the formula

then from the (nonstandard) triangle inequality we have

(say). On the other hand, has the Lipschitz continuity property

and so in particular we see that

for some Lipschitz continuous function . If we then let be the set where , one can check that differs from by a set of Loeb outer measure , and hence does so also. Sending to zero, we see (from the converse claim) that is a Cauchy sequence in and thus converges in for some Lebesgue measurable . The sets then converge in Loeb outer measure to , giving the claim.

Thanks to the Lebesgue differentiation theorem, the conditional expectation of a bounded Loeb-measurable function can be expressed (as a function on , defined -a.e.) as

By the abstract ergodic theorem from the previous post, one can also view this conditional expectation as the element in the closed convex hull of the shifts , of minimal norm. In particular, we obtain a form of the von Neumann ergodic theorem in this context: the averages for converge (as a net, rather than a sequence) in to .

If is (the standard part of) an internal function, that is to say the ultralimit of a sequence of finitary bounded functions, one can view the measurable function as a limit of the that is analogous to the “graphons” that emerge as limits of graphs (see e.g. the recent text of Lovasz on graph limits). Indeed, the measurable function is related to the discrete functions by the formula

for all , where is the nonprincipal ultrafilter used to define the nonstandard universe. In particular, from the Arzela-Ascoli diagonalisation argument there is a subsequence such that

thus is the asymptotic density function of the . For instance, if is the indicator function of a randomly chosen subset of , then the asymptotic density function would equal (almost everywhere, at least).

I’m continuing to look into understanding the ergodic theory of actions, as I believe this may allow one to apply ergodic theory methods to the “single-scale” or “non-asymptotic” setting (in which one averages only over scales comparable to a large parameter , rather than the traditional asymptotic approach of letting the scale go to infinity). I’m planning some further posts in this direction, though this is still a work in progress.

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space , and is a vector in that Hilbert space, then one has

in the strong topology, where is the -invariant subspace of , and is the orthogonal projection to . (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if is a countable amenable group acting on a Hilbert space by unitary transformations , and is a vector in that Hilbert space, then one has

for any Folner sequence of , where is the -invariant subspace. Thus one can interpret as a certain average of elements of the orbit of .

I recently discovered that there is a simple variant of this ergodic theorem that holds even when the group is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem)Let be an arbitrary group acting unitarily on a Hilbert space , and let be a vector in . Then is the element in the closed convex hull of of minimal norm, and is also the unique element of in this closed convex hull.

*Proof:* As the closed convex hull of is closed, convex, and non-empty in a Hilbert space, it is a classical fact (see e.g. Proposition 1 of this previous post) that it has a unique element of minimal norm. If for some , then the midpoint of and would be in the closed convex hull and be of smaller norm, a contradiction; thus is -invariant. To finish the first claim, it suffices to show that is orthogonal to every element of . But if this were not the case for some such , we would have for all , and thus on taking convex hulls , a contradiction.

Finally, since is orthogonal to , the same is true for for any in the closed convex hull of , and this gives the second claim.

This result is due to Alaoglu and Birkhoff. It implies the amenable ergodic theorem (1); indeed, given any , Theorem 1 implies that there is a finite convex combination of shifts of which lies within (in the norm) to . By the triangle inequality, all the averages also lie within of , but by the Folner property this implies that the averages are eventually within (say) of , giving the claim.

It turns out to be possible to use Theorem 1 as a substitute for the mean ergodic theorem in a number of contexts, thus removing the need for an amenability hypothesis. Here is a basic application:

Corollary 2 (Relative orthogonality)Let be a group acting unitarily on a Hilbert space , and let be a -invariant subspace of . Then and are relatively orthogonal over their common subspace , that is to say the restrictions of and to the orthogonal complement of are orthogonal to each other.

*Proof:* By Theorem 1, we have for all , and the claim follows. (Thanks to Gergely Harcos for this short argument.)

Now we give a more advanced application of Theorem 1, to establish some “Mackey theory” over arbitrary groups . Define a *-system* to be a probability space together with a measure-preserving action of on ; this gives an action of on , which by abuse of notation we also call :

(In this post we follow the usual convention of defining the spaces by quotienting out by almost everywhere equivalence.) We say that a -system is *ergodic* if consists only of the constants.

(A technical point: the theory becomes slightly cleaner if we interpret our measure spaces abstractly (or “pointlessly“), removing the underlying space and quotienting by the -ideal of null sets, and considering maps such as only on this quotient -algebra (or on the associated von Neumann algebra or Hilbert space ). However, we will stick with the more traditional setting of classical probability spaces here to keep the notation familiar, but with the understanding that many of the statements below should be understood modulo null sets.)

A *factor* of a -system is another -system together with a *factor map* which commutes with the -action (thus for all ) and respects the measure in the sense that for all . For instance, the *-invariant factor* , formed by restricting to the invariant algebra , is a factor of . (This factor is the first factor in an important hierachy, the next element of which is the *Kronecker factor* , but we will not discuss higher elements of this hierarchy further here.) If is a factor of , we refer to as an *extension* of .

From Corollary 2 we have

Corollary 3 (Relative independence)Let be a -system for a group , and let be a factor of . Then and are relatively independent over their common factor , in the sense that the spaces and are relatively orthogonal over when all these spaces are embedded into .

This has a simple consequence regarding the product of two -systems and , in the case when the action is trivial:

Lemma 4If are two -systems, with the action of on trivial, then is isomorphic to in the obvious fashion.

This lemma is immediate for countable , since for a -invariant function , one can ensure that holds simultaneously for all outside of a null set, but is a little trickier for uncountable .

*Proof:* It is clear that is a factor of . To obtain the reverse inclusion, suppose that it fails, thus there is a non-zero which is orthogonal to . In particular, we have orthogonal to for any . Since lies in , we conclude from Corollary 3 (viewing as a factor of ) that is also orthogonal to . Since is an arbitrary element of , we conclude that is orthogonal to and in particular is orthogonal to itself, a contradiction. (Thanks to Gergely Harcos for this argument.)

Now we discuss the notion of a group extension.

Definition 5 (Group extension)Let be an arbitrary group, let be a -system, and let be a compact metrisable group. A-extensionof is an extension whose underlying space is (with the product of and the Borel -algebra on ), the factor map is , and the shift maps are given bywhere for each , is a measurable map (known as the

cocycleassociated to the -extension ).

An important special case of a -extension arises when the measure is the product of with the Haar measure on . In this case, also has a -action that commutes with the -action, making a -system. More generally, could be the product of with the Haar measure of some closed subgroup of , with taking values in ; then is now a system. In this latter case we will call *-uniform*.

If is a -extension of and is a measurable map, we can define the *gauge transform* of to be the -extension of whose measure is the pushforward of under the map , and whose cocycles are given by the formula

It is easy to see that is a -extension that is isomorphic to as a -extension of ; we will refer to and as *equivalent* systems, and as *cohomologous* to . We then have the following fundamental result of Mackey and of Zimmer:

Theorem 6 (Mackey-Zimmer theorem)Let be an arbitrary group, let be an ergodic -system, and let be a compact metrisable group. Then every ergodic -extension of is equivalent to an -uniform extension of for some closed subgroup of .

This theorem is usually stated for amenable groups , but by using Theorem 1 (or more precisely, Corollary 3) the result is in fact also valid for arbitrary groups; we give the proof below the fold. (In the usual formulations of the theorem, and are also required to be Lebesgue spaces, or at least standard Borel, but again with our abstract approach here, such hypotheses will be unnecessary.) Among other things, this theorem plays an important role in the Furstenberg-Zimmer structural theory of measure-preserving systems (as well as subsequent refinements of this theory by Host and Kra); see this previous blog post for some relevant discussion. One can obtain similar descriptions of non-ergodic extensions via the ergodic decomposition, but the result becomes more complicated to state, and we will not do so here.

Given two unit vectors in a real inner product space, one can define the *correlation* between these vectors to be their inner product , or in more geometric terms, the cosine of the angle subtended by and . By the Cauchy-Schwarz inequality, this is a quantity between and , with the extreme positive correlation occurring when are identical, the extreme negative correlation occurring when are diametrically opposite, and the zero correlation occurring when are orthogonal. This notion is closely related to the notion of correlation between two non-constant square-integrable real-valued random variables , which is the same as the correlation between two unit vectors lying in the Hilbert space of square-integrable random variables, with being the normalisation of defined by subtracting off the mean and then dividing by the standard deviation of , and similarly for and .

One can also define correlation for complex (Hermitian) inner product spaces by taking the real part of the complex inner product to recover a real inner product.

While reading the (highly recommended) recent popular maths book “How not to be wrong“, by my friend and co-author Jordan Ellenberg, I came across the (important) point that correlation is not necessarily transitive: if correlates with , and correlates with , then this does not imply that correlates with . A simple geometric example is provided by the three unit vectors

in the Euclidean plane : and have a positive correlation of , as does and , but and are not correlated with each other. Or: for a typical undergraduate course, it is generally true that good exam scores are correlated with a deep understanding of the course material, and memorising from flash cards are correlated with good exam scores, but this does not imply that memorising flash cards is correlated with deep understanding of the course material.

However, there are at least two situations in which some partial version of transitivity of correlation can be recovered. The first is in the “99%” regime in which the correlations are *very* close to : if are unit vectors such that is *very* highly correlated with , and is *very* highly correlated with , then this *does* imply that is very highly correlated with . Indeed, from the identity

(and similarly for and ) and the triangle inequality

Thus, for instance, if and , then . This is of course closely related to (though slightly weaker than) the triangle inequality for angles:

Remark 1(Thanks to Andrew Granville for conversations leading to this observation.) The inequality (1) also holds for sub-unit vectors, i.e. vectors with . This comes by extending in directions orthogonal to all three original vectors and to each other in order to make them unit vectors, enlarging the ambient Hilbert space if necessary. More concretely, one can apply (1) to the unit vectorsin .

But even in the “” regime in which correlations are very weak, there is still a version of transitivity of correlation, known as the *van der Corput lemma*, which basically asserts that if a unit vector is correlated with *many* unit vectors , then many of the pairs will then be correlated with each other. Indeed, from the Cauchy-Schwarz inequality

Thus, for instance, if for at least values of , then must be at least , which implies that for at least pairs . Or as another example: if a random variable exhibits at least positive correlation with other random variables , then if , at least two distinct must have positive correlation with each other (although this argument does not tell you *which* pair are so correlated). Thus one can view this inequality as a sort of `pigeonhole principle” for correlation.

A similar argument (multiplying each by an appropriate sign ) shows the related van der Corput inequality

and this inequality is also true for complex inner product spaces. (Also, the do not need to be unit vectors for this inequality to hold.)

Geometrically, the picture is this: if positively correlates with all of the , then the are all squashed into a somewhat narrow cone centred at . The cone is still wide enough to allow a few pairs to be orthogonal (or even negatively correlated) with each other, but (when is large enough) it is not wide enough to allow *all* of the to be so widely separated. Remarkably, the bound here does not depend on the dimension of the ambient inner product space; while increasing the number of dimensions should in principle add more “room” to the cone, this effect is counteracted by the fact that in high dimensions, almost all pairs of vectors are close to orthogonal, and the exceptional pairs that are even weakly correlated to each other become exponentially rare. (See this previous blog post for some related discussion; in particular, Lemma 2 from that post is closely related to the van der Corput inequality presented here.)

A particularly common special case of the van der Corput inequality arises when is a unit vector fixed by some unitary operator , and the are shifts of a single unit vector . In this case, the inner products are all equal, and we arrive at the useful van der Corput inequality

(In fact, one can even remove the absolute values from the right-hand side, by using (2) instead of (4).) Thus, to show that has negligible correlation with , it suffices to show that the shifts of have negligible correlation with each other.

Here is a basic application of the van der Corput inequality:

Proposition 1 (Weyl equidistribution estimate)Let be a polynomial with at least one non-constant coefficient irrational. Then one haswhere .

Note that this assertion implies the more general assertion

for any non-zero integer (simply by replacing by ), which by the Weyl equidistribution criterion is equivalent to the sequence being asymptotically equidistributed in .

*Proof:* We induct on the degree of the polynomial , which must be at least one. If is equal to one, the claim is easily established from the geometric series formula, so suppose that and that the claim has already been proven for . If the top coefficient of is rational, say , then by partitioning the natural numbers into residue classes modulo , we see that the claim follows from the induction hypothesis; so we may assume that the top coefficient is irrational.

In order to use the van der Corput inequality as stated above (i.e. in the formalism of inner product spaces) we will need a non-principal ultrafilter (see e.g this previous blog post for basic theory of ultrafilters); we leave it as an exercise to the reader to figure out how to present the argument below without the use of ultrafilters (or similar devices, such as Banach limits). The ultrafilter defines an inner product on bounded complex sequences by setting

Strictly speaking, this inner product is only positive semi-definite rather than positive definite, but one can quotient out by the null vectors to obtain a positive-definite inner product. To establish the claim, it will suffice to show that

for every non-principal ultrafilter .

Note that the space of bounded sequences (modulo null vectors) admits a shift , defined by

This shift becomes unitary once we quotient out by null vectors, and the constant sequence is clearly a unit vector that is invariant with respect to the shift. So by the van der Corput inequality, we have

for any . But we may rewrite . Then observe that if , is a polynomial of degree whose coefficient is irrational, so by induction hypothesis we have for . For we of course have , and so

for any . Letting , we obtain the claim.

Let be a quasiprojective variety defined over a finite field , thus for instance could be an affine variety

where is -dimensional affine space and are a finite collection of polynomials with coefficients in . Then one can define the set of -rational points, and more generally the set of -rational points for any , since can be viewed as a field extension of . Thus for instance in the affine case (1) we have

The Weil conjectures are concerned with understanding the number

of -rational points over a variety . The first of these conjectures was proven by Dwork, and can be phrased as follows.

Theorem 1 (Rationality of the zeta function)Let be a quasiprojective variety defined over a finite field , and let be given by (2). Then there exist a finite number of algebraic integers (known ascharacteristic valuesof ), such thatfor all .

After cancelling, we may of course assume that for any and , and then it is easy to see (as we will see below) that the become uniquely determined up to permutations of the and . These values are known as the *characteristic values* of . Since is a rational integer (i.e. an element of ) rather than merely an algebraic integer (i.e. an element of the ring of integers of the algebraic closure of ), we conclude from the above-mentioned uniqueness that the set of characteristic values are invariant with respect to the Galois group . To emphasise this Galois invariance, we will not fix a specific embedding of the algebraic numbers into the complex field , but work with all such embeddings simultaneously. (Thus, for instance, contains three cube roots of , but which of these is assigned to the complex numbers , , will depend on the choice of embedding .)

An equivalent way of phrasing Dwork’s theorem is that the (-form of the) zeta function

associated to (which is well defined as a formal power series in , at least) is equal to a rational function of (with the and being the poles and zeroes of respectively). Here, we use the formal exponential

Equivalently, the (-form of the) zeta-function is a meromorphic function on the complex numbers which is also periodic with period , and which has only finitely many poles and zeroes up to this periodicity.

Dwork’s argument relies primarily on -adic analysis – an analogue of complex analysis, but over an algebraically complete (and metrically complete) extension of the -adic field , rather than over the Archimedean complex numbers . The argument is quite effective, and in particular gives explicit upper bounds for the number of characteristic values in terms of the complexity of the variety ; for instance, in the affine case (1) with of degree , Bombieri used Dwork’s methods (in combination with Deligne’s theorem below) to obtain the bound , and a subsequent paper of Hooley established the slightly weaker bound purely from Dwork’s methods (a similar bound had also been pointed out in unpublished work of Dwork). In particular, one has bounds that are uniform in the field , which is an important fact for many analytic number theory applications.

These -adic arguments stand in contrast with Deligne’s resolution of the last (and deepest) of the Weil conjectures:

Theorem 2 (Riemann hypothesis)Let be a quasiprojective variety defined over a finite field , and let be a characteristic value of . Then there exists a natural number such that for every embedding , where denotes the usual absolute value on the complex numbers . (Informally: and all of its Galois conjugates have complex magnitude .)

To put it another way that closely resembles the classical Riemann hypothesis, all the zeroes and poles of the -form lie on the critical lines for . (See this previous blog post for further comparison of various instantiations of the Riemann hypothesis.) Whereas Dwork uses -adic analysis, Deligne uses the essentially orthogonal technique of ell-adic cohomology to establish his theorem. However, ell-adic methods can be used (via the Grothendieck-Lefschetz trace formula) to establish rationality, and conversely, in this paper of Kedlaya p-adic methods are used to establish the Riemann hypothesis. As pointed out by Kedlaya, the ell-adic methods are tied to the intrinsic geometry of (such as the structure of sheaves and covers over ), while the -adic methods are more tied to the *extrinsic* geometry of (how sits inside its ambient affine or projective space).

In this post, I would like to record my notes on Dwork’s proof of Theorem 1, drawing heavily on the expositions of Serre, Hooley, Koblitz, and others.

The basic strategy is to control the rational integers both in an “Archimedean” sense (embedding the rational integers inside the complex numbers with the usual norm ) as well as in the “-adic” sense, with the characteristic of (embedding the integers now in the “complexification” of the -adic numbers , which is equipped with a norm that we will recall later). (This is in contrast to the methods of ell-adic cohomology, in which one primarily works over an -adic field with .) The Archimedean control is trivial:

Proposition 3 (Archimedean control of )With as above, and any embedding , we havefor all and some independent of .

*Proof:* Since is a rational integer, is just . By decomposing into affine pieces, we may assume that is of the affine form (1), then we trivially have , and the claim follows.

Another way of thinking about this Archimedean control is that it guarantees that the zeta function can be defined holomorphically on the open disk in of radius centred at the origin.

The -adic control is significantly more difficult, and is the main component of Dwork’s argument:

Proposition 4 (-adic control of )With as above, and using an embedding (defined later) with the characteristic of , we can find for any real a finite number of elements such thatfor all .

Another way of thinking about this -adic control is that it guarantees that the zeta function can be defined *meromorphically* on the entire -adic complex field .

Proposition 4 is ostensibly much weaker than Theorem 1 because of (a) the error term of -adic magnitude at most ; (b) the fact that the number of potential characteristic values here may go to infinity as ; and (c) the potential characteristic values only exist inside the complexified -adics , rather than in the algebraic integers . However, it turns out that by combining -adic control on in Proposition 4 with the trivial control on in Proposition 3, one can obtain Theorem 1 by an elementary argument that does not use any further properties of (other than the obvious fact that the are rational integers), with the in Proposition 4 chosen to exceed the in Proposition 3. We give this argument (essentially due to Borel) below the fold.

The proof of Proposition 4 can be split into two pieces. The first piece, which can be viewed as the number-theoretic component of the proof, uses external descriptions of such as (1) to obtain the following decomposition of :

Proposition 5 (Decomposition of )With and as above, we can decompose as a finite linear combination (over the integers) of sequences , such that for each such sequence , the zeta functionsare entire in , by which we mean that

as .

This proposition will ultimately be a consequence of the properties of the Teichmuller lifting .

The second piece, which can be viewed as the “-adic complex analytic” component of the proof, relates the -adic entire nature of a zeta function with control on the associated sequence , and can be interpreted (after some manipulation) as a -adic version of the Weierstrass preparation theorem:

Proposition 6 (-adic Weierstrass preparation theorem)Let be a sequence in , such that the zeta functionis entire in . Then for any real , there exist a finite number of elements such that

for all and some .

Clearly, the combination of Proposition 5 and Proposition 6 (and the non-Archimedean nature of the norm) imply Proposition 4.

This is a blog version of a talk I recently gave at the IPAM workshop on “The Kakeya Problem, Restriction Problem, and Sum-product Theory”.

Note: the discussion here will be highly non-rigorous in nature, being extremely loose in particular with asymptotic notation and with the notion of dimension. Caveat emptor.

One of the most infamous unsolved problems at the intersection of geometric measure theory, incidence combinatorics, and real-variable harmonic analysis is the Kakeya set conjecture. We will focus on the following three-dimensional case of the conjecture, stated informally as follows:

Conjecture 1 (Kakeya conjecture)Let be a subset of that contains a unit line segment in every direction. Then .

This conjecture is not precisely formulated here, because we have not specified exactly what type of set is (e.g. measurable, Borel, compact, etc.) and what notion of dimension we are using. We will deliberately ignore these technical details in this post. It is slightly more convenient for us here to work with lines instead of unit line segments, so we work with the following slight variant of the conjecture (which is essentially equivalent):

Conjecture 2 (Kakeya conjecture, again)Let be a family of lines in that meet and contain a line in each direction. Let be the union of the restriction to of every line in . Then .

As the space of all directions in is two-dimensional, we thus see that is an (at least) two-dimensional subset of the four-dimensional space of lines in (actually, it lies in a compact subset of this space, since we have constrained the lines to meet ). One could then ask if this is the only property of that is needed to establish the Kakeya conjecture, that is to say if any subset of which contains a two-dimensional family of lines (restricted to , and meeting ) is necessarily three-dimensional. Here we have an easy counterexample, namely a plane in (passing through the origin), which contains a two-dimensional collection of lines. However, we can exclude this case by adding an additional axiom, leading to what one might call a “strong” Kakeya conjecture:

Conjecture 3 (Strong Kakeya conjecture)Let be a two-dimensional family of lines in that meet , and assume theWolff axiomthat no (affine) plane contains more than a one-dimensional family of lines in . Let be the union of the restriction of every line in . Then .

Actually, to make things work out we need a more quantitative version of the Wolff axiom in which we constrain the metric entropy (and not just dimension) of lines that lie *close* to a plane, rather than exactly *on* the plane. However, for the informal discussion here we will ignore these technical details. Families of lines that lie in different directions will obey the Wolff axiom, but the converse is not true in general.

In 1995, Wolff established the important lower bound (for various notions of dimension, e.g. Hausdorff dimension) for sets in Conjecture 3 (and hence also for the other forms of the Kakeya problem). However, there is a key obstruction to going beyond the barrier, coming from the possible existence of *half-dimensional (approximate) subfields* of the reals . To explain this problem, it easiest to first discuss the complex version of the strong Kakeya conjecture, in which all relevant (real) dimensions are doubled:

Conjecture 4 (Strong Kakeya conjecture over )Let be a four (real) dimensional family of complex lines in that meet the unit ball in , and assume theWolff axiomthat no four (real) dimensional (affine) subspace contains more than a two (real) dimensional family of complex lines in . Let be the union of the restriction of every complex line in . Then has real dimension .

The argument of Wolff can be adapted to the complex case to show that all sets occuring in Conjecture 4 have real dimension at least . Unfortunately, this is sharp, due to the following fundamental counterexample:

Proposition 5 (Heisenberg group counterexample)Let be the Heisenberg groupand let be the family of complex lines

with and . Then is a five (real) dimensional subset of that contains every line in the four (real) dimensional set ; however each four real dimensional (affine) subspace contains at most a two (real) dimensional set of lines in . In particular, the strong Kakeya conjecture over the complex numbers is false.

This proposition is proven by a routine computation, which we omit here. The group structure on is given by the group law

giving the structure of a -step simply-connected nilpotent Lie group, isomorphic to the usual Heisenberg group over . Note that while the Heisenberg group is a counterexample to the complex strong Kakeya conjecture, it is not a counterexample to the complex form of the original Kakeya conjecture, because the complex lines in the Heisenberg counterexample do not point in distinct directions, but instead only point in a three (real) dimensional subset of the four (real) dimensional space of available directions for complex lines. For instance, one has the one real-dimensional family of parallel lines

with ; multiplying this family of lines on the right by a group element in gives other families of parallel lines, which in fact sweep out all of .

The Heisenberg counterexample ultimately arises from the “half-dimensional” (and hence degree two) subfield of , which induces an involution which can then be used to define the Heisenberg group through the formula

Analogous Heisenberg counterexamples can also be constructed if one works over finite fields that contain a “half-dimensional” subfield ; we leave the details to the interested reader. Morally speaking, if in turn contained a subfield of dimension (or even a subring or “approximate subring”), then one ought to be able to use this field to generate a counterexample to the strong Kakeya conjecture over the reals. Fortunately, such subfields do not exist; this was a conjecture of Erdos and Volkmann that was proven by Edgar and Miller, and more quantitatively by Bourgain (answering a question of Nets Katz and myself). However, this fact is not entirely trivial to prove, being a key example of the sum-product phenomenon.

We thus see that to go beyond the dimension bound of Wolff for the 3D Kakeya problem over the reals, one must do at least one of two things:

- (a) Exploit the distinct directions of the lines in in a way that goes beyond the Wolff axiom; or
- (b) Exploit the fact that does not contain half-dimensional subfields (or more generally, intermediate-dimensional approximate subrings).

(The situation is more complicated in higher dimensions, as there are more obstructions than the Heisenberg group; for instance, in four dimensions quadric surfaces are an important obstruction, as discussed in this paper of mine.)

Various partial or complete results on the Kakeya problem over various fields have been obtained through route (a) or route (b). For instance, in 2000, Nets Katz, Izabella Laba and myself used route (a) to improve Wolff’s lower bound of for Kakeya sets very slightly to (for a weak notion of dimension, namely upper Minkowski dimension). In 2004, Bourgain, Katz, and myself established a sum-product estimate which (among other things) ruled out approximate intermediate-dimensional subrings of , and then pursued route (b) to obtain a corresponding improvement to the Kakeya conjecture over finite fields of prime order. The analogous (discretised) sum-product estimate over the reals was established by Bourgain in 2003, which in principle would allow one to extend the result of Katz, Laba and myself to the strong Kakeya setting, but this has not been carried out in the literature. Finally, in 2009, Dvir used route (a) and introduced the polynomial method (as discussed previously here) to completely settle the Kakeya conjecture in finite fields.

Below the fold, I present a heuristic argument of Nets Katz and myself, which in principle would use route (b) to establish the full (strong) Kakeya conjecture. In broad terms, the strategy is as follows:

- Assume that the (strong) Kakeya conjecture fails, so that there are sets of the form in Conjecture 3 of dimension for some . Assume that is “optimal”, in the sense that is as large as possible.
- Use the optimality of (and suitable non-isotropic rescalings) to establish strong forms of standard structural properties expected of such sets , namely “stickiness”, “planiness”, “local graininess” and “global graininess” (we will roughly describe these properties below). Heuristically, these properties are constraining to “behave like” a putative Heisenberg group counterexample.
- By playing all these structural properties off of each other, show that can be parameterised locally by a one-dimensional set which generates a counterexample to Bourgain’s sum-product theorem. This contradiction establishes the Kakeya conjecture.

Nets and I have had an informal version of argument for many years, but were never able to make a satisfactory theorem (or even a partial Kakeya result) out of it, because we could not rigorously establish anywhere near enough of the necessary structural properties (stickiness, planiness, etc.) on the optimal set for a large number of reasons (one of which being that we did not have a good notion of dimension that did everything that we wished to demand of it). However, there is beginning to be movement in these directions (e.g. in this recent result of Guth using the polynomial method obtaining a weak version of local graininess on certain Kakeya sets). In view of this (and given that neither Nets or I have been actively working in this direction for some time now, due to many other projects), we’ve decided to distribute these ideas more widely than before, and in particular on this blog.

Roth’s theorem on arithmetic progressions asserts that every subset of the integers of positive upper density contains infinitely many arithmetic progressions of length three. There are many versions and variants of this theorem. Here is one of them:

Theorem 1 (Roth’s theorem)Let be a compact abelian group, with Haar probability measure , which is -divisible (i.e. the map is surjective) and let be a measurable subset of with for some . Then we havewhere denotes the bound for some depending only on .

This theorem is usually formulated in the case that is a finite abelian group of odd order (in which case the result is essentially due to Meshulam) or more specifically a cyclic group of odd order (in which case it is essentially due to Varnavides), but is also valid for the more general setting of -divisible compact abelian groups, as we shall shortly see. One can be more precise about the dependence of the implied constant on , but to keep the exposition simple we will work at the qualitative level here, without trying at all to get good quantitative bounds. The theorem is also true without the -divisibility hypothesis, but the proof we will discuss runs into some technical issues due to the degeneracy of the shift in that case.

We can deduce Theorem 1 from the following more general Khintchine-type statement. Let denote the Pontryagin dual of a compact abelian group , that is to say the set of all continuous homomorphisms from to the (additive) unit circle . Thus is a discrete abelian group, and functions have a Fourier transform defined by

If is -divisible, then is -torsion-free in the sense that the map is injective. For any finite set and any radius , define the *Bohr set*

where denotes the distance of to the nearest integer. We refer to the cardinality of as the *rank* of the Bohr set. We record a simple volume bound on Bohr sets:

Lemma 2 (Volume packing bound)Let be a compact abelian group with Haar probability measure . For any Bohr set , we have

*Proof:* We can cover the torus by translates of the cube . Then the sets form an cover of . But all of these sets lie in a translate of , and the claim then follows from the translation invariance of .

Given any Bohr set , we define a normalised “Lipschitz” cutoff function by the formula

where is the constant such that

thus

The function should be viewed as an -normalised “tent function” cutoff to . Note from Lemma 2 that

We then have the following sharper version of Theorem 1:

Theorem 3 (Roth-Khintchine theorem)Let be a -divisible compact abelian group, with Haar probability measure , and let . Then for any measurable function , there exists a Bohr set with and such thatwhere denotes the convolution operation

A variant of this result (expressed in the language of ergodic theory) appears in this paper of Bergelson, Host, and Kra; a combinatorial version of the Bergelson-Host-Kra result that is closer to Theorem 3 subsequently appeared in this paper of Ben Green and myself, but this theorem arguably appears implicitly in a much older paper of Bourgain. To see why Theorem 3 implies Theorem 1, we apply the theorem with and equal to a small multiple of to conclude that there is a Bohr set with and such that

But from (2) we have the pointwise bound , and Theorem 1 follows.

Below the fold, we give a short proof of Theorem 3, using an “energy pigeonholing” argument that essentially dates back to the 1986 paper of Bourgain mentioned previously (not to be confused with a later 1999 paper of Bourgain on Roth’s theorem that was highly influential, for instance in emphasising the importance of Bohr sets). The idea is to use the pigeonhole principle to choose the Bohr set to capture all the “large Fourier coefficients” of , but such that a certain “dilate” of does not capture much more Fourier energy of than itself. The bound (3) may then be obtained through elementary Fourier analysis, without much need to explicitly compute things like the Fourier transform of an indicator function of a Bohr set. (However, the bound obtained by this argument is going to be quite poor – of tower-exponential type.) To do this we perform a structural decomposition of into “structured”, “small”, and “highly pseudorandom” components, as is common in the subject (e.g. in this previous blog post), but even though we crucially need to retain non-negativity of one of the components in this decomposition, we can avoid recourse to conditional expectation with respect to a partition (or “factor”) of the space, using instead convolution with one of the considered above to achieve a similar effect.

Throughout this post, we will work only at the *formal* level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. (Rigorous justification of the conservation laws and other identities arising from the formal manipulations below can usually be established in an *a posteriori* fashion once the identities are in hand, without the need to rigorously justify the manipulations used to come up with these identities).

It is a remarkable fact in the theory of differential equations that many of the ordinary and partial differential equations that are of interest (particularly in geometric PDE, or PDE arising from mathematical physics) admit a variational formulation; thus, a collection of one or more fields on a domain taking values in a space will solve the differential equation of interest if and only if is a critical point to the functional

involving the fields and their first derivatives , where the Lagrangian is a function on the vector bundle over consisting of triples with , , and a linear transformation; we also usually keep the boundary data of fixed in case has a non-trivial boundary, although we will ignore these issues here. (We also ignore the possibility of having additional constraints imposed on and , which require the machinery of Lagrange multipliers to deal with, but which will only serve as a distraction for the current discussion.) It is common to use local coordinates to parameterise as and as , in which case can be viewed locally as a function on .

Example 1 (Geodesic flow)Take and to be a Riemannian manifold, which we will write locally in coordinates as with metric for . A geodesic is then a critical point (keeping fixed) of the energy functionalor in coordinates (ignoring coordinate patch issues, and using the usual summation conventions)

As discussed in this previous post, both the Euler equations for rigid body motion, and the Euler equations for incompressible inviscid flow, can be interpreted as geodesic flow (though in the latter case, one has to work

reallyformally, as the manifold is now infinite dimensional).More generally, if is itself a Riemannian manifold, which we write locally in coordinates as with metric for , then a harmonic map is a critical point of the energy functional

or in coordinates (again ignoring coordinate patch issues)

If we replace the Riemannian manifold by a Lorentzian manifold, such as Minkowski space , then the notion of a harmonic map is replaced by that of a wave map, which generalises the scalar wave equation (which corresponds to the case ).

Example 2 (-particle interactions)Take and ; then a function can be interpreted as a collection of trajectories in space, which we give a physical interpretation as the trajectories of particles. If we assign each particle a positive mass , and also introduce a potential energy function , then it turns out that Newton’s laws of motion in this context (with the force on the particle being given by the conservative force ) are equivalent to the trajectories being a critical point of the action functional

Formally, if is a critical point of a functional , this means that

whenever is a (smooth) deformation with (and with respecting whatever boundary conditions are appropriate). Interchanging the derivative and integral, we (formally, at least) arrive at

Write for the infinitesimal deformation of . By the chain rule, can be expressed in terms of . In coordinates, we have

where we parameterise by , and we use subscripts on to denote partial derivatives in the various coefficients. (One can of course work in a coordinate-free manner here if one really wants to, but the notation becomes a little cumbersome due to the need to carefully split up the tangent space of , and we will not do so here.) Thus we can view (2) as an integral identity that asserts the vanishing of a certain integral, whose integrand involves , where vanishes at the boundary but is otherwise unconstrained.

A general rule of thumb in PDE and calculus of variations is that whenever one has an integral identity of the form for some class of functions that vanishes on the boundary, then there must be an associated differential identity that justifies this integral identity through Stokes’ theorem. This rule of thumb helps explain why integration by parts is used so frequently in PDE to justify integral identities. The rule of thumb can fail when one is dealing with “global” or “cohomologically non-trivial” integral identities of a topological nature, such as the Gauss-Bonnet or Kazhdan-Warner identities, but is quite reliable for “local” or “cohomologically trivial” identities, such as those arising from calculus of variations.

In any case, if we apply this rule to (2), we expect that the integrand should be expressible as a spatial divergence. This is indeed the case:

Proposition 1(Formal) Let be a critical point of the functional defined in (1). Then for any deformation with , we havewhere is the vector field that is expressible in coordinates as

*Proof:* Comparing (4) with (3), we see that the claim is equivalent to the Euler-Lagrange equation

The same computation, together with an integration by parts, shows that (2) may be rewritten as

Since is unconstrained on the interior of , the claim (6) follows (at a formal level, at least).

Many variational problems also enjoy one-parameter continuous *symmetries*: given any field (not necessarily a critical point), one can place that field in a one-parameter family with , such that

for all ; in particular,

which can be written as (2) as before. Applying the previous rule of thumb, we thus expect another divergence identity

whenever arises from a continuous one-parameter symmetry. This expectation is indeed the case in many examples. For instance, if the spatial domain is the Euclidean space , and the Lagrangian (when expressed in coordinates) has no direct dependence on the spatial variable , thus

then we obtain translation symmetries

for , where is the standard basis for . For a fixed , the left-hand side of (7) then becomes

where . Another common type of symmetry is a *pointwise* symmetry, in which

for all , in which case (7) clearly holds with .

If we subtract (4) from (7), we obtain the celebrated theorem of Noether linking symmetries with conservation laws:

Theorem 2 (Noether’s theorem)Suppose that is a critical point of the functional (1), and let be a one-parameter continuous symmetry with . Let be the vector field in (5), and let be the vector field in (7). Then we have the pointwise conservation law

In particular, for one-dimensional variational problems, in which , we have the conservation law for all (assuming of course that is connected and contains ).

Noether’s theorem gives a systematic way to locate conservation laws for solutions to variational problems. For instance, if and the Lagrangian has no explicit time dependence, thus

then by using the time translation symmetry , we have

as discussed previously, whereas we have , and hence by (5)

and so Noether’s theorem gives conservation of the *Hamiltonian*

For instance, for geodesic flow, the Hamiltonian works out to be

so we see that the speed of the geodesic is conserved over time.

For pointwise symmetries (9), vanishes, and so Noether’s theorem simplifies to ; in the one-dimensional case , we thus see from (5) that the quantity

is conserved in time. For instance, for the -particle system in Example 2, if we have the translation invariance

for all , then we have the pointwise translation symmetry

for all , and some , in which case , and the conserved quantity (11) becomes

as was arbitrary, this establishes conservation of the *total momentum*

Similarly, if we have the rotation invariance

for any and , then we have the pointwise rotation symmetry

for any skew-symmetric real matrix , in which case , and the conserved quantity (11) becomes

since is an arbitrary skew-symmetric matrix, this establishes conservation of the *total angular momentum*

Below the fold, I will describe how Noether’s theorem can be used to locate all of the conserved quantities for the Euler equations of inviscid fluid flow, discussed in this previous post, by interpreting that flow as geodesic flow in an infinite dimensional manifold.

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