In Lecture 11, we studied compact measure-preserving systems – those systems (X, {\mathcal X}, \mu, T) in which every function f \in L^2(X, {\mathcal X}, \mu) was almost periodic, which meant that their orbit \{ T^n f: n \in {\Bbb Z}\} was precompact in the L^2(X, {\mathcal X}, \mu) topology. Among other things, we were able to easily establish the Furstenberg recurrence theorem (Theorem 1 from Lecture 11) for such systems.

In this lecture, we generalise these results to a “relative” or “conditional” setting, in which we study systems which are compact relative to some factor (Y, {\mathcal Y}, \nu, S) of (X, {\mathcal X}, \mu, T). Such systems are to compact systems as isometric extensions are to isometric systems in topological dynamics. The main result we establish here is that the Furstenberg recurrence theorem holds for such compact extensions whenever the theorem holds for the base. The proof is essentially the same as in the compact case; the main new trick is to not to work in the Hilbert spaces L^2(X,{\mathcal X},\mu) over the complex numbers, but rather in the Hilbert module L^2(X,{\mathcal X},\mu|Y, {\mathcal Y}, \nu) over the (commutative) von Neumann algebra L^\infty(Y,{\mathcal Y},\nu). (Modules are to rings as vector spaces are to fields.) Because of the compact nature of the extension, it turns out that results from topological dynamics (and in particular, van der Waerden’s theorem) can be exploited to good effect in this argument.

[Note: this operator-algebraic approach is not the only way to understand these extensions; one can also proceed by disintegrating \mu into fibre measures \mu_y for almost every y \in Y and working fibre by fibre. We will discuss the connection between the two approaches below.]

– Hilbert modules –

Let X = (X,{\mathcal X},\mu,T) be a measure-preserving system, and let \pi: X \to Y be a factor map, i.e. a morphism from X to another system Y = (Y, {\mathcal Y}, \nu, S). The algebra L^\infty(Y) can be viewed (using \pi) as a subalgebra of L^\infty(X); indeed, it is isomorphic to L^\infty(X, \pi^{\#}({\mathcal Y}), \mu), where \pi^{\#}({\mathcal Y}) := \{ \pi^{-1}(E): E \in {\mathcal Y} \} is the pullback of {\mathcal Y} by \pi.

Example 1. Throughout these notes we shall use the skew shift as our running example. Thus, in this example, X = ({\Bbb R}/{\Bbb Z})^2 with shift T: (y,z) \mapsto (y+\alpha,z+y) for some fixed \alpha (which can be either rational or irrational), Y = {\Bbb R}/{\Bbb Z} with shift S: y \mapsto y+\alpha, with factor map \pi: (y,z) \mapsto y. In this case, L^\infty(Y) can be thought of (modulo equivalence on null sets, of course) as the space of bounded functions on ({\Bbb R}/{\Bbb Z})^2 which depend only on the first variable. \diamond

Example 2. Another (rather trivial) example is when the factor system Y is simply a point. In this case, L^\infty(Y) is the space of constants and can be identified with {\Bbb C}. At the opposite extreme, another example is when Y is equal to X. It is instructive to see how all of the concepts behave in each of these two extreme cases, as well as the typical intermediate case presented in Example 1. \diamond

The idea here will be to try to “relativise” the machinery of Hilbert spaces over {\Bbb C} to that of Hilbert modules over L^\infty(Y). Roughly speaking, all concepts which used to be complex or real-valued (e.g. inner products, norms, coefficients, etc.) will now take values in the algebra L^\infty(Y). The following table depicts the various concepts that will be relativised:

Absolute / unconditional Relative / conditional
Constants {\Bbb C} Factor-measurable functions L^\infty(Y)
Expectation {\Bbb E} f = \int_X f\ d\mu \in {\Bbb C} Conditional expectation {\Bbb E}(f|Y) \in L^\infty(Y)
Inner product \langle f, g \rangle_{L^2(X)} = {\Bbb E} f \overline{g} Conditional inner product \langle f, g \rangle_{L^2(X|Y)} = {\Bbb E}(f \overline g|Y)
Hilbert space L^2(X) Hilbert module L^2(X|Y)
Finite-dimensional subspace \{\sum_{j=1}^d c_j f_j: c_1,\ldots,c_d \in {\Bbb C}\} Finitely generated module \{ \sum_{j=1}^d c_j f_j: c_1,\ldots,c_d \in L^\infty(Y)\}
Almost periodic function Conditionally almost periodic function
Compact system Compact extension
Hilbert-Schmidt operator Conditionally Hilbert-Schmidt operator
Weakly mixing function Conditionally weakly mixing function
Weakly mixing system Weakly mixing extension

(The last few notions in this table will be covered in the next lecture, rather than this one.)

Remark 1. In information-theoretic terms, one can view Y as representing all the observables in the system X that have already been “measured” in some sense, so that it is now permissible to allow one’s “constants” to depend on that data, and only study the remaining information present in X conditioning on the observed values in Y. Note though that once we activate the shift map T, the data in Y will similarly shift (by S), and so the various fibres of \pi can interact with each other in a non-trivial manner, so one should take some caution in applying information-theoretic intuition to this setting. \diamond

We have already seen that the factor Y induces a sub-\sigma-algebra \pi^{\#}({\mathcal Y}) of {\mathcal X}. We therefore have a conditional expectation map f \mapsto {\Bbb E}(f|Y) defined for all absolutely integrable f by the formula

{\Bbb E}(f|Y) := {\Bbb E}(f|\pi^{\#}({\mathcal Y})). (1)

In general, this expectation only lies in L^1(Y), though for the functions we shall eventually study, the expectation will always lie in L^\infty(Y) when needed.

As stated in the table, conditional expectation will play the role in the conditional setting that the unconditional expectation {\Bbb E} f = \int_X f\ d\mu plays in the unconditional setting. Note though that the conditional expectation takes values in the algebra L^\infty(Y) rather than in the complex numbers. We recall that conditional expectation is linear over this algebra, thus

{\Bbb E}(cf + dg|Y) = c {\Bbb E}(f|Y) + d {\Bbb E}(g|Y) (2)

for all absolutely integrable f, g and all c, d \in L^\infty(Y).

Example 3. Continuing Example 1, we see that for any absolutely integrable f on ({\Bbb R}/{\Bbb Z})^2, we have {\Bbb E}(f|Y)(y,z) = \int_{{\Bbb R}/{\Bbb Z}} f(y,z')\ dz' almost everywhere. \diamond
Let L^2(X|Y) be the space of all f \in L^2(X,{\mathcal X},\mu) such that the conditional norm

\|f\|_{L^2(X|Y)} := {\Bbb E}(|f|^2 | Y)^{1/2} (3)

lies in L^\infty(Y) (rather than merely in L^2(Y), which it does automatically). Thus for instance we have the inclusions

L^\infty(X) \subset L^2(X|Y) \subset L^2(X). (4)

The space L^2(X|Y) is easily seen to be a vector space over {\Bbb C}, and moreover (thanks to (2)) is a module over L^\infty(Y).

Exercise 1. If we introduce the inner product

\langle f, g \rangle_{L^2(X|Y)} := {\Bbb E}(f \overline{g}|Y) (5)

(which, initially, is only in L^1(Y)), establish the pointwise Cauchy-Schwarz inequality

|\langle f, g \rangle_{L^2(X|Y)}| \leq \|f\|_{L^2(X|Y)} \|g\|_{L^2(X|Y)} (6)

almost everywhere. In particular, the inner product lies in L^\infty(Y). (Hint: repeat the standard proof of the Cauchy-Schwarz inequality verbatim, but with L^\infty(Y) playing the role of the constants {\Bbb C}.) \diamond

Example 4. Continuing Examples 1 and 3, L^2(X|Y) consists (modulo null set equivalence) of all measurable functions f(y,z) such that \|f\|_{L^2(X|Y)} = (\int_{{\Bbb R}/{\Bbb Z}} |f(y,z)|^2\ dz)^{1/2} is bounded a.e. in y, with the relative inner product

\langle f, g \rangle_{L^2(X|Y)}(y) := \int_{{\Bbb R}/{\Bbb Z}} f(y,z) \overline{g(y,z)}\ dz (7)

defined a.e. in y. Observe in this case that the relative Cauchy-Schwarz inequality (6) follows easily from the standard Cauchy-Schwarz inequality. \diamond

Exercise 2. Show that the function f \mapsto \| \|f\|_{L^2(X|Y)} \|_{L^\infty(Y)} is a norm on L^2(X|Y), which turns that space into a Banach space. (Hint: you may need to “relativise” one of the standard proofs that L^2(X) is complete. You may also want to start with the skew shift example to build some intuition.) Because of this completeness, we refer to L^2(X|Y) as a Hilbert module over L^\infty(Y). \diamond

As \pi is a morphism, one can easily check the intertwining relationship

{\Bbb E}(T^n f|Y) = S^n {\Bbb E}(f|Y) (7)

for all f \in L^1(X) and integers n. As a consequence we see that the map T (and all of its powers) preserves the space L^2(X|Y), and furthermore is conditionally unitary in the sense that

\langle T^n f, T^n g \rangle_{L^2(X|Y)} = S^n \langle f, g \rangle_{L^2(X|Y)} (8)

for all f, g \in L^2(X|Y) and integers n.

In the Hilbert space L^2(X) one can create finite dimensional subspaces \{ c_1 f_1 + \ldots + c_d f_d: c_1,\ldots,c_d \in {\Bbb C} \} for any f_1,\ldots,f_d \in L^2(X). Inside such subspaces we can create the bounded finite-dimensional zonotopes \{ c_1 f_1 + \ldots + c_d f_d: c_1,\ldots,c_d \in {\Bbb C}, |c_1|, \ldots,|c_d| \leq 1 \}. Observe (from the Heine-Borel theorem) that a subset E of L^2(X) is precompact if and only if it can be approximated by finite-dimensional zonotopes in the sense that for every \varepsilon > 0, there exists a finite-dimensional zonotope Z of L^2(X) such that E lies within the \varepsilon neighbourhood of Z.

Remark 2. There is nothing special about zonotopes here; just about any family of bounded finite-dimensional objects would suffice for this purpose. In fact, it seems to be slightly better (for the purposes of quantitative analysis, and in particular in controlling the dependence on dimension d) to work instead with octahedra, in which the constraint |c_1|,\ldots,|c_d| \leq 1 is replaced by |c_1| + \ldots + |c_d| = 1; see this paper of mine in which this perspective is used. \diamond

Inspired by this, let us make some definitions. A finitely generated module of L^2(X|Y) is any submodule of L^2(X|Y) of the form \{c_1 f_1 + \ldots + c_d f_d: c_1,\ldots,c_d \in L^\infty(Y)\}, where f_1,\ldots,f_d \in L^2(X|Y). Inside such a module we can define a finitely generated module zonotope \{c_1 f_1 + \ldots + c_d f_d: c_1,\ldots,c_d \in L^\infty(Y); \|c_1\|_{L^\infty(Y)}, \ldots, \|c_d\|_{L^\infty(Y)} \leq 1\}.

Definition 1. A subset E of L^2(X|Y) is said to be conditionally precompact if for every \varepsilon > 0, there exists a finitely generated module zonotope Z of L^2(X|Y) such that E lies within the \varepsilon-neighbourhood of Z (using the norm from Exercise 2).

A function f \in L^2(X|Y) is said to be conditionally almost periodic if its orbit \{ T^n f: n \in {\Bbb Z} \} is conditionally precompact.

A function f \in L^2(X|Y) is said to be conditionally almost periodic in measure if every \varepsilon > 0 there exists a set E in Y of measure at most \varepsilon such that f1_{E^c} is conditionally almost periodic.

The system X is said to be a compact extension of Y if every function in L^2(X|Y) is conditionally almost periodic in measure.

Example 5. Any bounded subset of L^\infty(Y) is conditionally precompact (though note that it need not be precompact in the topological sense, using the topology from Exercise 2). In particular, every function in L^\infty(Y) is conditionally almost periodic. \diamond

Example 6. Every system is a compact extension of itself. A system is a compact extension of a point if and only if it is a compact system. \diamond

Example 7. Consider the skew shift (Examples 1, 3, 4), and consider the orbit of the function f(y,z) := e^{2\pi i z}. A computation shows that

T^n f(y,z) = e^{2\pi i \frac{-n(-n-1)}{2} \alpha} e^{-2\pi i n y} f (9)

which reveals (for \alpha irrational) that f is not almost periodic in the unconditional sense. However, observe that all the shifts T^n f lie in the zonotope \{ c f: c \in L^\infty(Y), \|c\|_{L^\infty(Y)} \leq 1 \} generated by a single generator f, and so f is conditionally almost periodic. \diamond

Exercise 3. Consider the skew shift (Examples 1,3,4,7). Show that a sequence f_n \in L^\infty(X) is conditionally precompact if and only if the sequences f_n(y,\cdot) \in L^\infty({\Bbb R}/{\Bbb Z}) are precompact in L^2({\Bbb R}/{\Bbb Z}) (with the usual Lebesgue measure) for almost every y. \diamond

Exercise 4. Show that the space of conditionally almost periodic functions in L^2(X|Y) is a shift-invariant L^\infty(Y) module, i.e. it is closed under addition, under multiplication by elements of L^\infty(Y), and under powers T^n of the shift operator. \diamond

Exercise 5. Consider the skew shift (Examples 1,3,4,7 and Exercise 3) with \alpha irrational, and let f \in L^2(X|Y) be the function defined by setting f(y,z) := e^{2\pi i n z} whenever n \geq 1 and y \in (1/(n+1),1/n]. Show that f is conditionally almost periodic in measure, but not conditionally almost periodic. Thus the two notions can be distinct even for bounded functions (a subtlety that does not arise in the unconditional setting). \diamond

Exercise 6. Let {\mathcal Z}_{X|Y} denote the collection of all measurable sets E in X such that 1_E is conditionally almost periodic in measure. Show that {\mathcal Z}_{X|Y} is a shift-invariant sub-\sigma-algebra of {\mathcal X} that contains \pi^{\#} {\mathcal Y}, and that a function f \in L^2(X|Y) is conditionally almost periodic in measure if and only if it is {\mathcal Z}-measurable. (In particular, (X, {\mathcal Z}_{X|Y}, \mu, T) is the maximal compact extension of Y.) [Hint: you may need to truncate the generators f_1,\ldots,f_d of various module zonotopes to be in L^\infty(X) rather than L^2(X|Y).] \diamond

Exercise 7. Show that the skew shift (Examples 1, 3, 4, 7 and Exercises 3,5) is a compact extension of the circle shift. (Hint: Use Example 7 and Exercise 6. Alternatively, approximate a function on the skew torus by its vertical Fourier expansions. For each fixed horizontal coordinate y, the partial sums of these vertical Fourier series converge (in the vertical L^2 sense) to the original function, pointwise in y. Now apply Egorov’s theorem.) \diamond

Exercise 8. Show that each of the iterated skew shifts (Exercise 8 from Lecture 9) are compact extensions of the preceding skew shift. \diamond

Exercise 9. Let (Y, {\mathcal Y}, \nu, S) be a measure-preserving system, let G be a compact metrisable group with a closed subgroup H, let \sigma: Y \to G be measurable, and let Y \times_\sigma G/H be the extension of Y with underlying space Y \times G/H , with measure equal to the product of \nu and Haar measure, and shift map T: (y,\zeta) \mapsto (Sy, \sigma(y) \zeta). Show that Y \times_\sigma G/H is a compact extension of Y. \diamond

– Multiple recurrence for compact extensions –

Let us say that a measure-preserving system (X, {\mathcal X}, \mu, T) obeys the uniform multiple recurrence (UMR) property if the conclusion of the Furstenberg multiple recurrence theorem holds for this system, thus for all k \geq 1 and all non-negative f \in L^\infty(X) with \int_X f\ d\mu > 0, we have

\liminf_{N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} \int_X f T^n f \ldots T^{(k-1) n} f\ d\mu > 0. (10)

Thus in Lecture 11 we showed that all compact systems obey UMR, and in Lecture 12 we showed that all weakly mixing systems obey UMR. The Furstenberg multiple recurrence theorem asserts, of course, that all measure-preserving systems obey UMR.

We now establish an important further step (and, in many ways, the key step) towards proving that theorem:

Theorem 1. Suppose that X = (X, {\mathcal X}, \mu, T) is a compact extension of Y = (Y, {\mathcal Y}, \nu, S). If Y obeys UMR, then so does X.

Note that the converse implication is trivial: if a system obeys UMR, then all of its factors automatically do also.

Proof of Theorem 1. Fix k \geq 1, and fix a non-negative function f \in L^\infty(X) with \int_X f\ d\mu > 0. Our objective is to show that (10) holds. As X is a compact extension, f is conditionally almost periodic in measure; by definition (and uniform integrability), this implies that f can be bounded from below by another conditionally almost periodic function which is non-negative with positive mean. Thus we may assume without loss of generality that f is conditionally almost periodic.

We may normalise \|f\|_{L^\infty(X)} = 1 and \int_X f\ d\mu = \delta for some 0 < \delta < 1. The reader may wish to follow this proof using the skew shift example as a guiding model.

Let \varepsilon > 0 be a small number (depending on k and \delta) to be chosen later. If we set E := \{ y \in Y: {\Bbb E}(f|Y) > \delta/2 \}, then E must have measure at least \delta/2.

Since f is almost periodic, we can find a finitely generated module zonotope \{ c_1 f_1 + \ldots + c_d f_d: \|c_1\|_{L^\infty(Y)},\ldots,\|c_d\|_{L^\infty(Y)} \leq 1 \} whose \varepsilon-neighbourhood contains the orbit of f. In other words, we have an identity of the form

T^n f = c_{1,n} f_1 + \ldots + c_{d,n} f_d + e_n (11)

for all n, where c_{1,n},\ldots,c_{d,n} \in L^\infty(Y) with norm at most 1, and e_n \in L^2(X,Y) is an error with \|e_n\|_{L^2(X|Y)} = O(\varepsilon) almost everywhere.

By splitting into real and imaginary parts (and doubling d if necessary) we may assume that the c_{j,n} are real-valued. By further duplication we can also assume that \|f_i\|_{L^2(X|Y)} \leq 1 for each i. By rounding off c_{j,n}(y) to the nearest multiple of \varepsilon/d for each y (and absorbing the error into the e_n term) we may assume that c_{j,n}(y) is always a multiple of \varepsilon/d. Thus each c_{j,n} only takes on O_{\varepsilon,d}(1) values.

Let K be a large integer (depending on k, d, \delta, \varepsilon) to be chosen later. Since the factor space Y obeys UMR, and E has positive measure in Y, we know that

\liminf_{N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} \int_Y 1_{E} T^n 1_{E} \ldots T^{(K-1) n}1_{E}\ d\nu > 0. (12)

In other words, there exists a constant c > 0 such that

\nu( \Omega_n ) > c (13)

for a set of n of positive lower density, where \Omega_n is the set

\Omega_n := E \cap T^n E \cap \ldots \cap T^{(K-1)n} E. (14)

Let n be as above. By definition of \Omega_n and E (and (8)), we see that

{\Bbb E}( T^{an} f | Y )(y) \geq \delta/2 (15)

for all y \in \Omega_n and 0 \leq a < K. Meanwhile, from (11) we have

\| T^{an} f - c_{1,an} f_1 - \ldots - c_{d,an} f_d \|_{L^2(X|Y)}(y) = O(\varepsilon) (16)

for all y \in \Omega_n and 0 \leq a < K.

Fix y. For each 0 \leq a < K, the d-tuple \vec c_{an}(y) := (c_{1,an}(y),\ldots,c_{d,an}(y)) ranges over a set of cardinality O_{d,\varepsilon}(1). One can view this as a colouring of \{0,\ldots,K-1\} into O_{d,\varepsilon}(1) colours. Applying van der Waerden’s theorem (Exercise 3 from Lecture 4), we can thus find (if K is sufficiently large depending on d, \varepsilon, k) an arithmetic progression a(y), a(y)+r(y), \ldots, a(y)+(k-1)r(y) in \{0,\ldots,K-1\} for each y such that

\vec c_{a(y)n}(y) = \vec c_{(a(y)+r(y)) n}(y) = \ldots = \vec c_{(a(y)+(k-1)r(y))n}(y). (17)

The quantities a(y) and r(y) can of course be chosen to be measurable in y. By the pigeonhole principle, we can thus find a subset \Omega'_n of \Omega_n of measure at least \sigma > 0 for some \sigma depending on c, K, d, \varepsilon but independent of n, and an arithmetic progression a, a+r, \ldots, a+(k-1)r in \{0,\ldots,K-1\} such that

\vec c_{an}(y) = \vec c_{(a+r) n}(y) = \ldots = \vec c_{(a+(k-1)r)n}(y) (18)

for all y \in \Omega'_n. (The quantities a and r can still depend on n, but this will not be of concern to us.)

Fix these values of a, r. From (16), (18) and the triangle inequality we see that

\| T^{(a+jr)n} f - T^{an} f \|_{L^2(X|Y)}(y) = O(\varepsilon) (19)
for all 1 \leq j \leq k and y \in \Omega'_n. Recalling that f was normalised to have L^\infty(X) norm 1, it is then not hard to conclude (by induction on k and the relative Cauchy-Schwarz inequality) that

\| T^{an} f T^{(a+r)n} f \ldots T^{(a+(k-1)r)n} f - (T^{an} f )^k \|_{L^2(X|Y)}(y) = O_k(\varepsilon) (20)

and thus (by another application of relative Cauchy-Schwarz)

{\Bbb E}( T^{an} f T^{(a+r)n} f \ldots T^{(a+(k-1)r)n} f )(y) \geq {\Bbb E}( (T^{an} f )^k | Y)(y) - O_k(\varepsilon). (21)

But from (15) we have

{\Bbb E}( T^{an} f | Y)(y) \geq \delta/2 (22)

and so by several more applications of relative Cauchy-Schwarz we have

{\Bbb E}( ( T^{an} f )^k|Y)(y) \geq c(k,\delta) > 0 (23)

for some positive quantity c(k,\delta). From (21), (23) we conclude that

{\Bbb E}( T^{an} f T^{(a+r)n} f \ldots T^{(a+(k-1)r)n} f )(y) \geq c(k,\delta)/2 (24)

for y \in \Omega'_n, if \varepsilon is small enough. Integrating this in y and using the shift-invariance we conclude that

\int_X f T^{nr} f \ldots T^{(k-1)nr} f\ d\mu \geq c(k,\delta) \sigma/2. (25)

The quantity r depends on n, but ranges between 1 and K-1, and so (by the non-negativity of f)

\sum_{s=1}^{K-1} \int_X f T^{ns} f \ldots T^{(k-1)ns} f\ d\mu \geq c(k,\delta) \sigma/2 (26)

for a set of n of positive lower density. Averaging this for n from 1 to N (say) one obtains (10) as desired. \Box

Thus for instance we have now established UMR for the skew shift as well as higher iterates of that shift, thanks to Exercises 7 and 8.

Remark 3. One can avoid the use of Hilbert modules, etc. by instead appealing to the theory of disintegration of measures (Theorem 4 from Lecture 9). We sketch the details as follows. First, one has to restrict attention to those spaces X which are regular, though an inspection of the Furstenberg correspondence principle (Lecture 10) shows that this is in fact automatic for the purposes of such tasks as proving Szemerédi’s theorem. Once one disintegrates \mu with respect to \nu, the situation now resembles the concrete example of the skew shift, with the fibre measures \mu_y playing the role of integration along vertical fibers \{ (y,z): z \in {\Bbb R}/{\Bbb Z} \}. It is then not difficult (and somewhat instructive) to convert the above proof to one using norms such as L^2(X, {\mathcal X}, mu_y) rather than the module norm L^2(X|Y). We leave the details to the reader (who can also get them from Furstenberg’s book). \diamond

Remark 4. It is an intriguing question as to whether there is any interesting non-commutative extension of the above theory, in which the underlying von Neumann algebra L^\infty(Y, {\mathcal Y},\nu) is replaced by a non-commutative von Neumann algebra. While some of the theory seems to extend relatively easily, there does appear to be some genuine difficulties with other parts of the theory, particularly those involving multiple products such as f T^n f T^{2n} f. \diamond

Remark 5. Just as ergodic compact systems can be described as group rotation systems (Kronecker systems), it turns out that ergodic compact extensions can be described as (inverse limits of) group quotient extensions, somewhat analogously to Lemma 2 from Lecture 6. Roughly speaking, the idea is to first use some spectral theory to approximate conditionally almost periodic functions by conditionally quasiperiodic functions – those functions whose orbit lies on a finitely generated module zonotope (as opposed to merely being close to one). One can then use the generators of that zonotope as a basis from which to build the group quotient extension, and then use some further trickery to make the group consistent across all fibres. The precise machinery for this is known as Mackey theory; it is of particular importance in the deeper structural theory of dynamical systems, but we will not describe it in detail here, instead referring the reader to the papers of Furstenberg and of Zimmer. \diamond

[Update, Mar 5: Exercise added.]

[Update, Mar 12: Definition of conditionally almost periodic measure tweaked (to an equivalent definition) in order to simplify the proof of Theorem 1.]