In these notes we quickly review the basics of abstract measure theory and integration theory, which was covered in the previous course but will of course be relied upon in the current course. This is only a brief summary of the material; of course, one should consult a real analysis text for the full details of the theory.
– Measurable spaces –
Ideally, measure theory on a space X should be able to assign a measure (or “volume”, or “mass”, etc.) to every set in X. Unfortunately, due to paradoxes such as the Banach-Tarski paradox, many natural notions of measure (e.g. Lebesgue measure) cannot be applied to measure all subsets of X; instead, one must restrict attention to certain measurable subsets of X. This turns out to suffice for most applications; for instance, just about any “non-pathological” subset of Euclidean space that one actually encounters will be Lebesgue measurable (as a general rule of thumb, any set which does not rely on the axiom of choice in its construction will be measurable).
To formalise this abstractly, we use
Definition 1. A measurable space is a set X, together with a collection of subsets of X which form a -algebra, thus contains the empty set and X, and is closed under countable intersections, countable unions, and complements. A subset of X is said to be measurable with respect to the measurable space if it lies in .
A function from one measurable space to another is said to be measurable if for all .
Remark 2. The notion of a measurable space (and of a measurable function) is superficially similar to that of a topological space (and of a continuous function); the topology contains and X just as the -algebra does, but is now closed under arbitrary unions and finite intersections, rather than countable unions, countable intersections, and complements. The two categories are linked to each other by the Borel algebra construction, see Example 2 below.
Example 1. We say that one -algebra on a set X is coarser than another (or that is finer than ) if (or equivalently, if the identity map from to is measurable); thus every set which is measurable in the coarse space is also measurable in the fine space. The coarsest -algebra on a set X is the trivial -algebra , while the finest is the discrete -algebra .
Example 2. The intersection of an arbitrary family of -algebras on X is another -algebra on X. Because of this, given any collection of sets on X we can define the -algebra generated by , defined to be the intersection of all the -algebras containing , or equivalently the coarsest algebra for which all sets in are measurable. (This intersection is non-vacuous, since it will always involve the discrete -algebra .) In particular, the open sets of a topological space generate a -algebra, known as the Borel -algebra of that space.
We can also define the join of any family of -algebras on X by the formula
For instance, the Lebesgue -algebra of Lebesgue measurable sets on a Euclidean space is the join of the Borel -algebra and of the algebra of null sets and their complements (also called co-null sets).
Exercise 1. A function from one topological space to another is said to be Borel measurable if it is measurable once X and Y are equipped with their respective Borel -algebras. Show that every continuous function is Borel measurable. (The converse statement, of course, is very far from being true; for instance, the pointwise limit of a sequence of measurable functions, if it exists, is also measurable, whereas the analogous claim for continuous functions is completely false.)
Remark 3. A function is said to be Lebesgue measurable if it is measurable from (with the Lebesgue -algebra) to (with the Borel -algebra), or equivalently if is Lebesgue measurable for every open ball B in . Note the asymmetry between Lebesgue and Borel here; in particular, the composition of two Lebesgue measurable functions need not be Lebesgue measurable.
Example 3. Given a function from a set X to a measurable space , we can define the pullback of to be the -algebra ; this is the coarsest structure on X that makes f measurable. For instance, the pullback of the Borel -algebra from [0,1] to under the map consists of all sets of the form , where is Borel-measurable.
More generally, given a family of functions into measurable spaces , we can form the -algebra generated by the ; this is the coarsest structure on X that makes all the simultaneously measurable.
Remark 4. In probability theory and information theory, the functions in Example 3 can be interpreted as observables, and the -algebra generated by these observables thus captures mathematically the concept of observable information. For instance, given a time parameter t, one might define the -algebra generated by all observables for some random process (e.g. Brownian motion) that can be made at time t or earlier; this endows the underlying event space X with an uncountable increasing family of -algebras.
Example 4. If E is a subset of a measurable space , the pullback of under the inclusion map is called the restriction of to E and is denoted . Thus, for instance, we can restrict the Borel and Lebesgue -algebras on a Euclidean space to any subset of such a space.
Exercise 2. Let M be an n-dimensional manifold, and let be an atlas of coordinate charts for M, where is an open cover of M and are open subsets of . Show that the Borel -algebra on M is the unique -algebra whose restriction to each is the pullback via of the restriction of the Borel -algebra of to .
Example 5. A function into some index set A will partition X into level sets for ; conversely, every partition of X arises from at least one function f in this manner (one can just take f to be the map from points in X to the partition cell that that point lies in). Given such an f, we call the -algebra the -algebra generated by the partition; a set is measurable with respect to this structure if and only if it is the union of some sub-collection of cells of the partition.
Exercise 3. Show that a -algebra on a finite set X necessarily arises from a partition as in Example 5, and furthermore the partition is unique (up to relabeling). Thus in the finite world, -algebras are essentially the same concept as partitions.
Example 6. Let be a family of measurable spaces, then the Cartesian product has canonical projection maps for each . The product -algebra is defined as the -algebra on generated by the as in Example 3.
Exercise 4. Let be an at most countable family of second countable topological spaces. Show that the Borel -algebra of the product space (with the product topology) is equal to the product of the Borel -algebras of the factor spaces. In particular, the Borel -algebra on is the product of n copies of the Borel -algebra on . (The claim can fail when the countability hypotheses are dropped, though in most applications in analysis, these hypotheses are satisfied.) We caution however that the Lebesgue -algebra on is not the product of n copies of the one-dimensional Lebesgue -algebra, as it contains some additional null sets; however, it is the completion of that product.
Exercise 5. Let and be measurable spaces. Show that if E is measurable with respect to , then for every , the set is measurable in , and similarly for every , the set is measurable in . Thus, sections of Borel-measurable sets are again Borel-measurable. (The same is not true for Lebesgue-measurable sets.)
– Measure spaces –
Now we endow measurable spaces with a measure, turning them into measure spaces.
Definition 2 (Measures) A (non-negative) measure on a measurable space is a function such that , and such that we have the countable additivity property whenever are disjoint measurable sets. We refer to the triplet as a measure space.
A measure space is finite if ; it is a probability space if (and then we call a probability measure). It is -finite if X can be covered by countably many sets of finite measure.
A measurable set E is a null set if . A property on points x in X is said to hold for almost every (or almost surely, for probability spaces) if it holds outside of a null set. We abbreviate almost every and almost surely as a.e. and a.s. respectively. The complement of a null set is said to be a co-null set or to have full measure.
Example 7. (Dirac measures) Given any measurable space and a point , we can define the Dirac measure (or Dirac mass) to be the measure such that when and otherwise. This is a probability measure.
Example 8. (Counting measure) Given any measurable space , we define counting measure by defining to be the cardinality |E| of E when E is finite, or otherwise. This measure is finite when X is finite, and -finite when X is at most countable. If X is also finite, we can define normalised counting measure ; this is a probability measure, also known as uniform probability measure on X (especially if we give X the discrete -algebra).
Example 9. Any finite non-negative linear combination of measures is again a measure; any finite convex combination of probability measures is again a probability measure.
Example 10. If is a measurable map from one measurable space to another , and is a measure on , we can define the push-forward by the formula ; this is a measure on . Thus, for instance, for all .
We record some basic properties of measures of sets:
Exercise 6. Let be a measure space.
- (Monotonicity) If are measurable sets, then . (In particular, any measurable subset of a null set is again a null set.)
- (Countable subadditivity) If are a countable sequence of measurable sets, then . (Of course, one also has subadditivity for finite sequences.) In particular, any countable union of null sets is again a null set.
- (Monotone convergence for sets) If are measurable, then .
- (Dominated convergence for sets) If are measurable, and is finite, then . Show that the claim can fail if is infinite.
Exercise 7. A measure space is said to be complete if every subset of a null set is measurable (and is thus again a null set). Show that every measure space has a unique minimal complete refinement , known as the completion of , and that a set is measurable in if and only if it is equal almost everywhere to a measurable set in . (The completion of the Borel -algebra with respect to Lebesgue measure is known as the Lebesgue -algebra.)
A powerful way to construct measures on -algebras is to first construct them on a smaller Boolean algebra that generates , and then extend them via the following result:
Theorem 1. (Carathéodory’s extension theorem, special case) Let be a measurable space, and let be a Boolean algebra (i.e. closed under finite unions, intersections, and complements) that generates . Let be a function such that
- If are disjoint and , then .
Then can be extended to a measure on , which we shall also call .
Remark 5. The conditions 1,2 in the above theorem are clearly necessary if has any hope to be extended to a measure on . Thus this theorem gives a necessary and sufficient condition for a function on a Boolean algebra to be extended to a measure. The extension can easily be shown to be unique when X is -finite.
Proof. (sketch) Define the outer measure of any set as the infimum of , where ranges over all coverings of E by elements in . It is not hard to see that if agrees with on , so it will suffice to show that it is a measure on .
It is easy to check that is monotone and countably subadditive (as in parts 1,2 of Exercise 5) on all of , and assigns zero to ; thus it is an outer measure in the abstract sense. But we need to show countable additivity on . The key is to first show the related property
for all and . This can first be shown for , and then one observes that the class of E that obey (2) for all A is a -algebra; we leave this as a (moderately lengthy) exercise.
The identity (2) already shows that is finitely additive on ; combining this with countable subadditivity and monotonicity, we conclude that is countably additive, as required.
Exercise 8. Let the notation and hypotheses be as in Theorem 1. Show that given any and any set of finite measure, there exists a set which differs from E by a set of measure at most . If X is -finite, show that the hypothesis that E have finite measure can be removed. (Hint: first reduce to the case when X is finite, then show that the class of all E obeying this property is a -algebra.) Thus sets in the -algebra “almost” lie in the algebra ; this is an example of Littlewood’s first principle. The same statements of course apply for the completion of .
One can use Theorem 1 to construct Lebesgue measure on and on (taking to be, say, the algebra generated by half-open intervals or boxes), although the verification of hypothesis 2 of Theorem 2 turns out to be somewhat delicate, even in the one-dimensional case. But one can at least get the higher-dimensional Lebesgue measure from the one-dimensional one by the product measure construction:
Exercise 9. Let be a finite collection of measure spaces, and let be the product measurable space . Show that there exists a unique measure on this space such that for all . The measure is referred to as the product measure of the and is denoted .
Exercise 10. Let E be a Lebesgue measurable subset of . and let m be Lebesgue measure. Establish the inner regularity property
and the outer regularity property
– Integration –
Now we define integration on a measure space .
Definition 3. (Integration) Let be a measure space.
- If is a non-negative simple function (i.e. a measurable function that only takes on finitely many values ), we define the integral of f to be (with the convention that ). In particular, if is the indicator function of a measurable set A, then .
- If is a non-negative measurable function, we define the integral to be the supremum of , where g ranges over all simple functions bounded between 0 and f.
- If is a measurable function, whose positive and negative parts , have finite integral, we say that f is absolutely integrable and define .
- If is a measurable function with real and imaginary parts absolutely integrable, we say that f is absolutely integrable and define .
We will sometimes show the variable of integration, e.g. writing for , for sake of clarity.
The following results are standard, and the proofs are omitted:
Theorem 2. (Standard facts about integration) Let be a measure space.
- All the above integration notions are compatible with each other; for instance, if f is both non-negative and absolutely integrable, then definitions 2 and 3 (and 4) agree.
- The functional is linear over for simple functions or non-negative functions, is linear over for real-valued absolutely integrable functions, and linear over for complex-valued absolutely integrable functions. In particular, the set of (real or complex) absolutely integrable functions on is a (real or complex) vector space.
- A complex-valued measurable function is absolutely integrable if and only if , in which case we have the triangle inequality . Of course, the same claim holds for real-valued measurable functions.
- If is non-negative, then , with equality holding if and only if f = 0 a.e.
- If one modifies an absolutely integrable function on a set of measure zero, then the new function is also absolutely integrable, and has the same integral as the original function. Similarly, two non-negative functions that agree a.e. have the same integral. (Because of this, we can meaningfully integrate functions that are only defined almost everywhere.)
- If is absolutely integrable, then f is finite a.e., and vanishes outside of a -finite set.
- If is absolutely integrable, and then there exists a complex-valued simple function such that . (This is a manifestation of Littlewood’s second principle.)
- (Change of variables formula) If is a measurable map to another measurable space , and , then we have , in the sense that whenever one of the integrals is well defined, then the other is also, and equals the first.
It is also important to note that the Lebesgue integral on extends the more classical Riemann integral. As a consequence, many properties of the Riemann integral (e.g. change of variables formula with respect to smooth diffeomorphisms) are inherited by the Lebesgue integral, thanks to various limiting arguments.
We now recall the fundamental convergence theorems relating limits and integration: the first three are for non-negative functions, the last three are for absolutely integrable functions. They are ultimately derived from their namesakes in Exercise 5 and an approximation argument by simple functions, and the proofs are again omitted. (They are also closely related to each other, and are in fact largely equivalent.)
Theorem 3. (Convergence theorems) Let be a measure space.
- (Monotone convergence for sequences) If are measurable, then .
- (Monotone convergence for series) If are measurable, then .
- (Fatou’s lemma) If are measurable, then .
- (Dominated convergence for sequences) If are measurable functions converging pointwise a.e. to a limit f, and a.e. for some absolutely integrable , then .
- (Dominated convergence for series) If are measurable functions with , then is absolutely convergent for a.e. x and .
- (Egorov’s theorem) If are measurable functions converging pointwise a.e. to a limit f on a subset A of X of finite measure, and , then there exists a set of measure at most , outside of which converges uniformly to f in A. (This is a manifestation of Littlewood’s third principle.)
Remark 7. As a rule of thumb, if one does not have exact or approximate monotonicity or domination (where “approximate” means “up to an error whose norm goes to zero”), then one should not expect the integral of a limit to equal the limit of the integral in general; there is just too much room for oscillation.
Exercise 11. Let be an absolutely integrable function on a measure space . Show that f is uniformly integrable, in the sense that for every there exists such that whenever E is a measurable set of measure at most . (The property of uniform integrability becomes more interesting, of course when applied to a family of functions, rather than to a single function.)
With regard to product measures and integration, the fundamental theorem in this subject is
Theorem 4. (Fubini-Tonelli theorem) Let and be -finite measure spaces, with product space .
- (Tonelli theorem) If is measurable, then .
- (Fubini theorem) If is absolutely integrable, then we also have , with the inner integrals being absolutely integrable a.e. and the outer integrals all being absolutely integrable.
- If and are complete measure spaces, then the same claims hold with the product -algebra replaced by its completion.
Remark 8. The theorem fails for non--finite spaces, but virtually every measure space actually encountered in “hard analysis” applications will be -finite. (One should be cautious, however, with any space constructed using ultrafilters or the first uncountable ordinal.) It is also important that f obey some measurability in the product space; there exist non-measurable f for which the iterated integrals exist (and may or may not be equal to each other, depending on the properties of f and even on which axioms of set theory one chooses), but the product integral (of course) does not.
[Update, Jan 20: statement of Egorov’s theorem corrected.]