Does anyone know where I find the proof of the following version of Poincaré recurrence:
Let $(X, \mathcal{B}, \mu, T)$ be a measure-preserving system with $\mu(X)=1$ only assumed to be a finitely additive measure ($\mu(∅) = 0$ and $\mu(A ∪ B) = \mu(A) + \mu(B)$ for any disjoint elements $A$ and $B$), and let $A \in \mathcal{B}$ have $\mu(A) >0$. Then there is some positive $n \leq \dfrac{1}{\mu(A)} $ for which $\mu(A ∩ T^{−n}A) > 0$