Recall the Poincare recurrence theorem, which states that for a finite measure space $X$ and a measure preserving map T, if $\mu(A)>0$ then $\mu$-almost point in $A$ returns to $A$ (that is, the $T$ orbit intersects $A$).
To the best of my knowledge, if we add the requirement that $T$ is ergodic, the conclusion holds for almost every point in $X$ (rather than only on $A$).
This makes we wounder, is the reverse statement correct? That is, assuming that the map is ergodic, if there is a set such that almost every orbit (of points in $X$) meets said set, must it have positive measure?
This seems very intuitive to me, as for ergodic maps the amount of time an orbit spends in a set corresponds to its measure, but I couldnt prove it. I mainly tried using Birkhoff's theorem. I wouldnt be surprised if it follows directly from some equivallent definition of ergodicity.
Hint: Suppose $U \subset X$ has the property that almost every point in $X$ has an orbit that intersects $U$. Note that
$$V \subset\bigcup_{n=0}^\infty T^{-n}(U)$$
where $V$ is the set that contains all points whose orbit intersects $U$. The conclusion should be clear once you take the measure of both sides of the above containment. (Note that ergodicity is not required).
Hope this helps!