Ergodic partial sums, time averages, and the invariant L2 subspace

Let $(X,\mathcal{A},\mu ,T)$ be a measure-preserving system (Measure-preserving transformations and systems), and put $T^0={\mathrm{id}}_X$ and $T^{k+1}=T\circ T^k$. For a finite-valued measurable real or complex representative $f$ and an integer $n\ge 1$, its ergodic partial sum and ergodic time average are

$S_{n}f:={\sum }_{k=0}^{n-1}f\circ T^k,A_{n}f:=\frac{1}{n}{S}_{n}f\mathrm{.}$

These are initially pointwise formulas for a chosen representative. The companion proposition Ergodic averages are measurable, representative independent, and Lp contractive ↗ proves that, when $f$ represents an element of $L^p(\mu )$, the formulas are independent of the representative and define $L^p$ classes. Thus the notation does not hide a simultaneous choice of representatives.

Using the complex-$L^2$ convention of Complex Lp classes and Euclidean test-function conventions, define the invariant $L^2$ subspace

$\mathcal{M}:=\{[g]\in L^2(\mu ;\mathbb{C}):[g\circ T]=[g]\}=\{[g]:g\circ T=g\mu \text{-a.e.}\}\mathrm{.}$

This is the fixed space of composition by $T$. Membership in $\mathcal{M}$ means invariance almost everywhere; it does not mean that $g$ is constant. The strict and modulo-null invariant-set conventions are those of Strict and mod-null invariant sigma-algebras.