SCHWEITZER 1959 Problem 6

Let $T$ be a one-to-one mapping of the unit square $E$ of the plane into itself. Suppose that $T$ and $T^{-1}$ are measure-preserving (i.e. if $M \subseteq E$ is a measurable set, then $TM$ and $T^{-1}M$ are also measurable and $\mu (M)= \mu (TM)= \mu (T^{-1}M)$ , where $\mu$ denotes the Lebesgue measure) and, furthermore, that if $Tx \in N$ for almost all points $x$ of a measurable set $N \subseteq E$ , then either $n$ or $E \setminus N$ is of measure 0. Prove that, for any measurable set $A \subseteq E$ , with $\mu (A)>0$ , the function $n(x)$ defined by $$n(x)=\begin{cases} 0, \text{if} T^k x \notin A (k=1, 2, \dots), \min (k: T^k x \in A; k=1,2, \dots ) &\text{otherwise} \end{cases}$$ is measurable and $\int_{A}n(x) d\mu(x) =1$ (R. 18)