I am having some trouble understanding the following:
Suppose $T:X\to X$ is a measurable transformation with probability measure $\rho$.
If $Y\subset X$ with positive measure and $R:Y\to \mathbb{N}$ is such that $T^{R(y)}(y)\in Y$ for all $y\in Y$, we can define the induced transformation by $F=T^R:Y\to Y$.
Suppose that $R\in L^1(Y)$. Assume there exists a unique $F$-ergodic measure $\nu$ on $Y$ such that $\nu$ is absolutely continuous with respect to $\rho|_Y$.
Define the measure $\mu$ on $X$ by $\mu(B)=\frac{1}{\|R\|_1}\sum_{k=1}^\infty\sum_{i=0}^{k-1}\nu(T^{-i}B\cap Y_k)$, where $Y_k=\{x\in Y : R(x)=k\}$. I want to show that
1) $\mu$ is an ergodic $T$-invariant measure on $X$.
2) If $T$ is non-singular wrt $\rho$, then $\mu$ is absolutely continuous with respect to $\rho$.
For 1), I can show invariance by calculating $\mu(T^{-1}B)-\mu(B)=\sum_{k=1}^\infty\nu(T^{-k}B\cap Y_k)-\sum_{k=1}^\infty\nu(B\cap Y_k)=\nu(F^{-1}B)-\nu(B)=0$. However, I am struggling to show ergodicity and 2). Can someone help? Thanks!
Source: For a more precise formulation see page 36 of this - https://www.fc.up.pt/pessoas/jfalves/slides.pdf.