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Multiple ergodic averages / Close Reading

A standard model for finitely many fixed observables

This is the reduction to a standard model used in A §2.1 (p.3) and B §2.1 (pp.3–4). The direct finite-orbit transference in Unit 1 for C does not require this step.

Let $(X,\mathcal F,\mu,T)$ be an invertible, bimeasurable, probability-preserving system on an arbitrary probability space, and fix $f_1,\ldots,f_d\in L^\infty(\mu)$ . No standardness assumption is made on the original probability space.

After normalizing the functions, choose measurable representatives satisfying $|f_j|\le1$ everywhere. Two representatives differ only on a null set, and the discrepancies for finitely many functions over all integer iterates still lie in a countable union of null sets. More precisely, if $g_j$ are the chosen representatives, all discrepancies are contained in

Z=\bigcup_{j=1}^d\bigcup_{k\in\mathbb Z}T^{-k}\{f_j\ne g_j\}.

This set is measurable, null, and $T$ -invariant, so it does not affect the almost-everywhere conclusions under consideration.

Let $D=\{z\in\mathbb C:|z|\le1\}$ , and set

Y=D^{\{1,\ldots,d\}\times\mathbb Z}.

A countable product of compact metric spaces is again compact and metrizable. With its Borel probability measure, this is a standard Borel probability space. Define

\pi(x)_{j,k}=f_j(T^kx),\quad (Sy)_{j,k}=y_{j,k+1},\quad \nu=\pi_*\mu.

Every coordinate is measurable, so $\pi$ is measurable. The map $S$ is an invertible continuous shift, and coordinatewise we have

\pi\circ T=S\circ\pi.

For every Borel set $E\subseteq Y$ ,

\nu(S^{-1}E) =\mu\bigl(\pi^{-1}(S^{-1}E)\bigr) =\mu\bigl(T^{-1}(\pi^{-1}E)\bigr) =\nu(E).

Thus $(Y,\nu,S)$ is probability-preserving.

Set $F_j(y)=y_{j,0}$ . Each $F_j$ is continuous and bounded, and for every integer $n$ ,

f_j(T^nx)=F_j(S^n\pi(x)).

For any fixed integer slopes $a_1,\ldots,a_d$ , the averages therefore satisfy the exact identity

\frac1N\sum_{n=1}^N\prod_{j=1}^df_j(T^{a_jn}x) = \left[\frac1N\sum_{n=1}^N\prod_{j=1}^dF_j\circ S^{a_jn}\right](\pi(x)).

If the averages on the right converge on a Borel set of full measure in $Y$ , its preimage has full measure in $X$ , so the averages on the left converge almost everywhere. The convergence set is indeed Borel: the Cauchy condition expresses it using countably many unions and intersections. No arbitrary nonmeasurable exceptional set is involved.

If an external theorem requires a complete standard probability space, one may complete $\nu$ before applying it. There is no need to assert, and one should not automatically infer, that $\pi$ remains measurable with respect to the completed target. The average functions above are continuous on $Y$ . Their convergence set, as well as the set on which they converge to a specified constant, already belongs to the original Borel $\sigma$ -algebra. It is enough to pull back this Borel set of full measure.

Mixing also passes to this factor. If the original system is mixing, then in $Y$ , any two Borel sets $E,G$ satisfy

\begin{aligned} \nu(E\cap S^{-n}G) &=\mu\bigl(\pi^{-1}E\cap T^{-n}(\pi^{-1}G)\bigr)\\ &\longrightarrow\mu(\pi^{-1}E)\mu(\pi^{-1}G) =\nu(E)\nu(G). \end{aligned}

Moreover, $\int_YF_j\,\mathrm{d}\nu=\int_Xf_j\,\mathrm{d}\mu$ , so a product-of-integrals limit pulls back unchanged.

This construction depends on the finite family of functions fixed in advance. It explains why many ergodic statements about fixed bounded observables pass from standard spaces to arbitrary probability spaces. It does not produce a single standard factor covering all $L^\infty$ functions of the original space, and it does not itself prove convergence of multiple averages.