Suppose $T:X\to X$ is an ergodic transformation with probability measure $\rho$.
If $Y\subset X$ with positive measure and $R:Y\to \mathbb{N}$ is such that $T^{R(y)}\in Y$ for all $y\in Y$, we can define the induced transformation by $F=T^R:Y\to Y$.
Is it true that $F$ is ergodic with respect to $\rho|_Y$? I feel like it should be, but whichever tool of ergodicity I use, I am thrown off by the return time function $R$.
If not, are there further conditions which guarantee ergodicity? e.g. $R$ being integrable.
Remark: I can show invariance, I am only interested in ergodicity!
Edit: it turns out (see the book Laws of Chaos by Gora and Boyarsky, Proposition 3.6.3) that ergodicity of T and F is equivalent if $\mu(X\setminus \bigcup_{n\ge 0}T^{-n}Y))=0$. Could anyone elaborate further on this condition? For example what guarantees such a condition is satisfied? Would $R$ being integrable be sufficient for this to hold?