Suppose that $(\Omega,\mathcal{A},\mu)$ is a $\sigma$-finite measure space of infinite measure and $T:\Omega\to\Omega$ a measure-preserving transformation with measurable inverse. Let be $\Omega_k\in \mathcal{A}$ an increasing sequence such that $\Omega_k\uparrow\Omega$ and $\mu(\Omega_k)<+\infty$ for all $k\in\mathbb{N}$.

** Question 1: ** Given a set $A\in\mathcal{A}$, such that $\mu(A)>0$, is it true that the set
$$E_k=\{\omega\in A; T^n(w)\notin A\ \forall n\in\mathbb{N}\ \text{and}\ T^{n_j}(w)\in \Omega_k \ \text{for some infinite sequence}\ (n_j(\omega)) \}$$
has zero measure ?

** Question 2: ** If $T$ is not invertible is $\mu(E_k)=0$, in general ?