In "Equivalence of measure preserving transformations" by Ornstein, Rudolph and Weiss, in Lemma 1.3, authors make a claim which can be summarised as follows:
Let $(X,\mathcal{B},\mu)$ be a probability space and let $T\colon X\to X$ be an invertible ergodic measure preserving transformation. If $f\colon X\to \mathbb{Z}$ is a measurable function with positive integral then, by the ergodic theorem, there exists an $N$ such that for all $n\geq N$ we have $$ \sum_{j<n} f\left(T^jx\right) > 0$$ for all $x$ in the set $B$ of measure at least 1/2. Authors claim that one can easily find a subset $A$ of $B$ such that $r_A(x)>N$ for all $x\in A$, where we define as usually $$ r_A(x) = \inf \left\{ n>0 : T^n x\in A \right\}.$$
I think I can see why this is true if $B$ contains subsets of arbitrarily small measure as one can show that $$ 1 = \int_A r_A\ d\mu \leq \mu(A)\sup r_A $$ so if we take smaller and smaller $\mu(A)$ then we will finally obtain points for which $r_A>N$ no matter how big $N$ we chose initially. But in general case the function $r_A$ may not attain arbitrarily large values. How can we then prove the authors' claim?