Let $T$ be a conservative measure preserving (non-invertible!) transformation of a measure space $(X, \mathscr{F}, m)$ with *infinite* measure $m$. Let $A \in \mathscr{F}$ be such that $X = \cup_{k=0}^\infty T^{-k} A \pmod{m}$ and $0<m(A)<\infty$. Then the first hitting time $\tau(x):= \inf\{k \ge 1: T^k x \in A\}$ is finite $m$-a.e. $x \in X$ and the induced transformation $T_A$ defined by $T_A(x) := T^{\tau(x)} x$ for $x \in A$ is measure preserving on $(A, \mathscr{F} \cap A, m|_A)$.

I am trying to prove that the invariant measure $m|_A$ of the induced transformation can be lifted back to $m$ as follows:
$$
m(B)=\int_A \sum_{k=0}^{\tau(x)-1} \mathbb 1(T^k x \in B) m|_A(dx), \qquad B \in \mathscr{F}.
$$
This is Lemma 1.5.4 of Aaronson's *Introduction to Infinite Ergodic Theory* but I do not understand the proof given, something is strange there. Is this statement actually true? Any hints or references please?