Let
- $(\Omega,\mathfrak A,\operatorname P)$ be a probability space
- $\Theta$ be a measurable map on $(\Omega,\mathfrak A)$ with $$\operatorname P\circ\:\Theta^{-1}=\operatorname P\tag1$$ and $$\mathcal I:=\left\{A\in\mathfrak A:A=\Theta^{-1}(A)\right\}$$
- $H$ be a $\mathbb R$-Hilbert space and $$\pi F:=\operatorname E\left[F\mid\mathcal I\right]$$ and $$A_nF:=\frac1n\sum_{i=0}^{n-1}F\circ\Theta^i\;\;\;\text{for }n\in\mathbb N$$ for $F\in\mathcal L^1(\operatorname P,H)$
I want to adopt this proof (step 4 on page 33) for the case $H=\mathbb R$ of the following claim: $$\left\|A_nF-\pi F\right\|_H\xrightarrow{n\to\infty}0\tag2\;\;\;\text{almost surely for all }F\in L^1(\operatorname P,H).$$
The idea of the given proof is the following: It is already known that there is a dense subspace $U$ (in the reference, $U=H_0+H_1$) of $L^1(\operatorname P,H)$ so that $(A_n-\pi)U\subseteq L^\infty(\operatorname P,H)$ for all $n\in\mathbb N$ and $$\left\|(A_n-\pi)F\right\|_{L^\infty(\operatorname P,\:H)}\xrightarrow{n\to\infty}0\;\;\;\text{for all }F\in U\tag3.$$ Moreover, it is already known that $$\left\|(A_n-\pi)F\right\|_{L^1(\operatorname P,\:H)}\xrightarrow{n\to\infty}0\;\;\;\text{for all }F\in L^1(\operatorname P,H)\tag4.$$
Let $F\in L^1(\operatorname P,H)$. The idea of the proof is to show that $(A_nF)_{n\in\mathbb N}$ is convergent outside a $\operatorname P$-null set. This is sufficient to conclude $(2)$, since by $(4)$ there is a subsequence of $(A_nF)_{n\in\mathbb N}$ which converges to $\pi F$ outside a $\operatorname P$-null set.
By the density assumption, there is a sequence $(F_k)_{k\in\mathbb N}\subseteq U$ so that $$\left\|F-F_k\right\|_{L^1(\operatorname P,\:H)}\xrightarrow{k\to\infty}0\tag5$$ and, by $(3)$, $$\left\|(A_n-\pi)F_k\right\|_H\le\left\|(A_n-\pi)F_k\right\|_{L^\infty(\operatorname P,\:H)}\xrightarrow{n\to\infty}0\;\;\;\text{on }\Omega\setminus N\tag6$$ for all $k\in\mathbb N$ for some $\operatorname P$-null set $N$.
Now, since $H=\mathbb R$, they argue by showing that \begin{equation}\begin{split}&\operatorname P\left[\limsup_{n\to\infty}A_nF-\liminf_{n\to\infty}A_nF>\varepsilon\right]\\&\;\;\;\;\;\;\;\;\;\;\;\;\le\operatorname P\left[\sup_{n\in\mathbb N}\left|A_n(F-F_k)\right|>\frac\varepsilon2\right]\end{split}\tag7\end{equation} for all $k\in\mathbb N$ and $\varepsilon>0$. Now the maximal ergodic theorem, $$\operatorname P\left[\sup_{n\in\mathbb N}\left\|A_nG\right\|>\varepsilon\right]\le\frac1\varepsilon\operatorname E\left[\left\|G\right\|_H\right]\;\;\;\text{for all }\varepsilon>0\tag8$$ for all $G\in L^1(\operatorname P,H)$, is used to conclude.
Can we still utilize $(8)$ in the general Hilbert space case?
Generally, a sequence $(x_n)_{n\in\mathbb N}$ in $H$ is convergent to $x$ if it weakly convergent to $x$ and $\left\|x_n\right\|_H\xrightarrow{n\to\infty}\left\|x\right\|_H$, but I don't see how we could use this here.