Hello mathematical fellows!
I've been reading from Krengel's book, "Ergodic Theorems" and in page 5 of the book just after the proof of von Neumann's mean ergodic theorem there is a proposition which tries to describe the limit as a conditional expectation, in detail the proposition states:
Proposition: If $\tau$ is an endomorphism (i.e measure preserving transformation) in $(\Omega,\mathscr{A},\mu)$ , and $\mu$ is $\sigma$-finite on the $\sigma$-algebra $\mathscr{B}$ of $\tau-$invariant sets, then, for any $f\in L_2(\Omega,\mathscr{A},\mu)$, the norm-limit $\overline{f}$ of $n^{-1}\sum_{k=0}^{n-1}f\circ \tau^k$ is given by $\overline{f}=\mathbb{E}(f\ |\ \mathscr{B})$ where $\mathbb{E}(f\ |\ \mathscr{B})$ denotes the conditional expectation of $f$ on the $\sigma-$algebra $\mathscr{B}$.
Note: $I_A$ is the indicator of $A$.
The author proves the proposition by saying word from word:
Proof: We may assume $\mu(\Omega)<\infty$.$\overline{f}$ is $\mathscr{B}$-measurable. For any $A\in \mathscr{A}$ we have $\int\limits_{A}f\circ \tau^k\ d\mu=\int\limits_{A}f\ d\mu$ because A is $\tau$-invariant and $\tau$ an endomorphism (preserves the measure $\mu$). Now $\int\limits_{A}f\ d\mu=< n^{-1} \sum\limits_{k=0}^{n-1} f\circ \tau^k,I_{A}>\to <\overline{f},I_A>=\int\limits_{A}\overline{f}\ d\mu$ because strong convergence implies weak convergence. $\square$
I understand the whole argument when $\mu$ is finite although at the $\sigma$-finite case i have two questions:
1)How can we now that for $f\in L_2(\Omega,\mathscr{A},\mu)$ the conditional expectation of $f$ on the $\sigma$-algebra $\mathscr{B}$ exists? (i.e an $\mathscr{B}$-measurable function $g $ s.t $\int_{A}f\ d\mu=\int_{A}g\ d\mu$ for every $A\in \mathscr{B}$. In the finite case the $\mathbb{E}(f\ |\ \mathscr{B})$ is well defined since $L_2\subseteq L_1$ and the existence follows from Radon-Nikodym theorem.
2)If we assume that $\mathbb{E}(f\ |\ \mathscr{B})$ exists how can we show that $\overline{f}=\mathbb{E}(f\ |\ \mathscr{B})$ when $\mu$ is infinite?
Thanks in advance!