Let $(X,B,\mu,T)$ be an invertible dynamicals system (i.e. $T^{-1}$ is measurable and exists almost everywhere)
Question 1:
is $T^{-1}$ also measure-preserving( $\mu(T(A)=\mu(A)$)?
Question 2:
if $f \in L_1(X)$
Then $\lim_{n\to\infty}1/n\sum_{i=0}^n{fT^i}=\lim_{n\to\infty}1/n\sum_{i=0}^n{fT^{-i}}$.
von Neumanns Theorem says, that the ergodic means converge to some invariant function g, but how can I see that these functions are equal? I guess somehow I have the intuition, because when I start at some value x (which orbit is not in the irregular set with measure 0), then it makes no difference in which direction ($T$ or $T^{-1}$) I start the iteration process for gaining the ergodic means...