$\lim_n \sup \frac{1}{n}\# \{0 ≤ j ≤ n − 1 : f^j(x) ∈ D \} > 0$

Question

Let $f:M \to M$ a measurable transformation and $\mu$ a invariant probability and $D \subset M $ a subset of positive measure. Prove that almost every point of $D$ passes a positive fraction of the time in $D$:

$$\limsup_n \dfrac{1}{n}\# \{0 ≤ j ≤ n − 1 : f^j(x) ∈ D \} > 0$$

for $μ$-almost every point $x ∈ D$.

I would try to prove that the set of $x$'s for which the given limit is zero has zero measure, but I do not know how to do it. Anyone have an idea how to get started?