Birkhoff limit for a stationary integrable process

The coordinate functional $f(z):=z_0$ is measurable on path space and belongs to $L^1({\mathbb{P}}_Y)$: by [F1] and the change-of-variables identity for the pushforward, ${\int }_{{\mathbb{R}}^{{\mathbb{N}}_0}}\mathrm{\mid }{z}_0\mathrm{\mid }d{\mathbb{P}}_Y=\mathbb{E}\mathrm{\mid }{Y}_0\mathrm{\mid }<\mathrm{\infty }$, and ${\mathbb{P}}_Y$ is a probability, hence finite and sigma-finite.

Let $A_{n}f:=n^{-1}{\sum }_{k<n}f\circ {\theta }^k$, so that $A_{n}f(z)=n^{-1}{\sum }_{k<n}{z}_k$. By [F3]–[F5] applied to the measure-preserving probability system $({\mathbb{R}}^{{\mathbb{N}}_0},{\mathbb{P}}_Y,\theta )$ and $f\in L^1({\mathbb{P}}_Y)$ there is $f^{\ast }\in L^1({\mathbb{P}}_Y)$ with: $A_{n}f\to f^{\ast }$ ${\mathbb{P}}_Y$-almost everywhere; $f^{\ast }$ is $\mathcal{I}$-measurable with ${\int }_E{f}^{\ast }d{\mathbb{P}}_Y={\int }_E{z}_{0}d{\mathbb{P}}_Y$ for every $E\in \mathcal{I}$; and $\parallel A_{n}f-f^{\ast }{\parallel }_{L^1({\mathbb{P}}_Y)}\to 0$.

By [F7] the function $f^{\ast }$ is a conditional-expectation version of $z_0$ given $\mathcal{I}$, that is, $f^{\ast }={\mathbb{E}}_{{\mathbb{P}}_Y}[z_0\mid \mathcal{I}]$ as an a.e. class; this is the unique a.e. class characterized by $\mathcal{I}$-measurability and the displayed integrals.

Pullback of the $L^1$ statement: for each $n$, ${\int }_{\mathrm{\Omega }}\mid \frac{1}{n}{\sum }_{k<n}{Y}_k-f^{\ast }\circ \mathrm{\Phi }\mid d\mathbb{P}={\int }_{{\mathbb{R}}^{{\mathbb{N}}_0}}\mathrm{\mid }{A}_{n}f-f^{\ast }\mathrm{\mid }d{\mathbb{P}}_Y$ by the pushforward identity [F1], and the right side tends to $0$ by step 2.1.

Pullback of the a.e. statement: $A_{n}f\circ \mathrm{\Phi }=\frac{1}{n}{\sum }_{k=0}^{n-1}{Y}_k$ as functions on $\mathrm{\Omega }$, and $\{z:A_{n}f(z)\to ̸f^{\ast }(z)\}$ is a ${\mathbb{P}}_Y$-null set; by [F1] its preimage under $\mathrm{\Phi }$ is a $\mathbb{P}$-null set, since $\mathbb{P}({\mathrm{\Phi }}^{-1}N)={\mathbb{P}}_Y(N)$. Hence $\frac{1}{n}{\sum }_{k<n}{Y}_k\to f^{\ast }\circ \mathrm{\Phi }$ almost surely.

If the canonical shift is ergodic for ${\mathbb{P}}_Y$ [F1, F2], then [F6] applies with $\mu ={\mathbb{P}}_Y$ and gives $A_{n}f\to \int z_{0}d{\mathbb{P}}_Y=\mathbb{E}{Y}_0$ ${\mathbb{P}}_Y$-a.e. and in $L^1({\mathbb{P}}_Y)$; pulling back as in steps 3.2 and 4.1 gives $\frac{1}{n}{\sum }_{k<n}{Y}_k\to \mathbb{E}{Y}_0$ almost surely and in $L^1(\mathbb{P})$.

Boundary and axiom cases: if $Y_0$ is a constant $c$ almost surely then all averages equal $c$, $\mathcal{I}$-measurability is automatic, and the ergodic conclusion is the same constant; if the process is ergodic but $Y_0$ is integrable with $\mathbb{E}{Y}_0=0$ the limit is the constant $0$, covered by step 5.1; the a.e. class of the limit is well defined because conditional-expectation versions are unique up to a.e. equality by [F7], and the theorem asserts convergence in two modes, not merely integrability of a limit; complex processes are handled by applying the real assertion to $\mathrm{Re}{Y}_n$ and $\mathrm{Im}{Y}_n$, both strictly stationary with finite first absolute moment, and recombining; and AC [A1] is used exactly through the identification [F4] and the uniqueness of the conditional-expectation class in [F7].