My question is on Ergodic Birkhoff Theorem, the state of this theorem is:
Birkhoff Ergodic Theorem: Let $(X, \mathcal{B}, \mu, T)$ be an ergodic measure-preserving dynamical system, where
$X$ is a measurable space,
$\mathcal{B}$ is a $\sigma$-algebra of subsets of $X$,
$\mu$ is a probability measure on $\mathcal{B}$,
$T: X \rightarrow X$ is a measure-preserving transformation, i.e., $\mu(T^{-1}(A)) = \mu(A)$ for all $A \in \mathcal{B}$,
and $\mu$ is ergodic, meaning that any $T$-invariant set has either measure $0$ or $1$.
Let $f: X \rightarrow \mathbb{R}$ be an integrable function with respect to $\mu$. Then, for almost every point $x \in X$, the time average of $f$ along the orbit of $x$ converges to the space average of $f$ with respect to $\mu$, i.e.,
\begin{align*}
\lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^{n-1} f(T^k(x)) = \int_X f \, d\mu \quad \text{for $\mu$-almost every } x \in X.
\end{align*}
This theorem is true for almost every point in the whole space $X$, Could we say this theorem is true for almost all points of $\text{supp} \mu$ ?
I was thinking of this in the way that $\mu(X \setminus \text{supp}\mu)=0$ so $\text{supp}\mu \overset{a.e}{=}X$
Best Answer
If something holds almost surely in $X$, it holds almost surely in every subspace of $X$.
Let $A$ be a measurable subset of $X$, $N$ a null set. $\mu(A\cap N)\le\mu(N)=0$ hence $A\cap N$ is also null; thus, $N$, or the restriction of the event $N$ to $A$, occurs almost never. So yes, the limit equation holds almost surely in the subspace $A$, in particular with $A=\mathrm{supp}\,\mu$.