Lifting back the induced invariant measure / general version of Kac's formula for occupation times

Question

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?

Answer 1

First of all, the question you ask has nothing to do with the finiteness (or not) of the measure $m$ as being measure preserving is a local property (it is formulated in terms of finite measure subsets). The relation between induced transformations and suspensions (these two operations are inverse to each other) is precisely the same no matter whether the state spaces are of finite measure or not.

You have already defined the induced transformation and formulated the fact that the restriction of an invariant measure is invariant under the induced transformation. Let me just mention that there is no need to specify anything concerning the measures of the state space $X$ and of the recurrent subset $A$; the only necessary assumption is that the hitting times are a.e. finite.

Conversely, given a measure preserving transformation $T:X\to X$ and an integer valued roof function $r$ one defines the suspension $\tilde T$ acting on the space $$ \tilde X = \{(x,n):1\le n \le r(x) \} $$ as $$ \tilde T(x,n) = \begin{cases} (x,n+1)\;,& n< r(x) \\ (Tx,1) \;, & n=r(x) \end{cases} $$ If $m$ is a measure on $X$, let $\tilde m$ denote the measure on $\tilde X$ obtained by integrating the counting measures on the fibers of the projection $\tilde X\to X$ with respect to $m$. If $m$ is $T$-invariant, then $\tilde m$ is $\tilde T$-invariant (the easiest way to check this is to look just at the subsets of $\tilde X$ of the form $A\times\{n\}$, where $A$ is a subset of $X$). This is precisely what you are asking about in the particular case when the roof function is the first return time.

PS Aaronson's book is written in a very dense formal style.

Answer 2

I was not able to find a reference and eventually proved the statement by myself, see Lemma A.3 (in the Appendix) of my paper https://arxiv.org/abs/1808.05010 The idea of the proof reminds that of Aaronson's.