I have a question about the invariant set in the ergodic theorem, I am wondering if anyone could give me some help or hint. In the measurable space (X, $\Sigma$) and consider a measurable self map T, then $A\in \Sigma$ is T-invariant if $$T^{-1}(A) = A$$
My question is that how to show that the set $$\left\{{\frac{\sum_{i=1}^{k} x\circ T^{i-1}}{k} converges}\right\} $$ is T-invariant, where x is a random variable on (X, $\Sigma$)
I encountered this problem when reading the ergodic theorem in Durret's Book and the ergodic theorem is based on $\frac{\sum_{i=1}^{k} x\circ T^{i-1}}{k}$, but I am not sure how to show that its convergence is a T-invariant set, since in the book, he did not mention it, maybe it is a trivial question, but I am wondering if anyone could deliver some help for this, thank you in advance!
Call $C$ this set and $M_k(\omega):=\frac 1k\sum_{i=0}^{k-1}x\circ T^i(\omega)$. Then $$M_k(T\omega)=\frac 1k\sum_{i=1}^kx\circ T^i(\omega)=\frac{k+1}k\left(M_{k+1}(\omega)-\frac{x(\omega)}{k+1}\right).$$ As for each $\omega$, $\frac{k+1}k\frac{x(\omega)}{k+1}\to 0$, the convergence of the sequence $(M_k(\omega),k\geqslant 1)$ is equivalent to the convergence of $(M_k(T(\omega)),k\geqslant 1)$, hence $C=T^{-1}(C)$.