Let $f:M\to M$, not necessarily invertible, $\mu$ an invariant probability measure for $f$ and $D\subset M$ a set with positive measure. Define $$D_0=\left\{ x\in D/ \lim_{n\to\infty}\frac{1}{n}\#\left\{0\leq j\leq n-1:\, f^j(x)\in D\right\}=0\right\}$$ It's clear that, for all $x\in D_0$ $$\lim_{n\to\infty}\frac{1}{n}\#\left\{0\leq j\leq n-1:\, f^j(x)\in D_0\right\}=0.$$ I need to prove that the above also happens for all $x\in M$.
I appreciate any suggestion.
If f be a ergodic map then the almost all the frequency of any $x\in X$ i.e. set ${x , f(x), f^2(x),...}$ lie in the set $D$ is $m(D)\neq 0$. So the measure of the set of elements that is have zero frequency in $D$ is zero.