Weakly wandering sets and invariant measures

Question

I am trying to show something but wherever I look in the literature, every paper seems to say 'obviously' or 'it is clear'. I'll now state the problem. Suppose we have a finite measure preserving dynamical system $(X,\mathcal{B},\mu,T)$. We say that a set of positive measure $W$ is weakly wondering if there is a sequence $n_i\to\infty$ such that $T^{n_i}W\cap T^{n_j}W=\emptyset$ for all $i\neq j$. Apparently, a finite measure preserving transformation $T$ cannot have a weakly wandering set. Everywhere in the literature says it is clear, but I can't see how. Could anyone please explain this? Thanks