Criterium in ergodic theory.

Question

Given a topological space $X$ with a probability measure $\mu$ and a continuous transformation $T:X \rightarrow X$ which preserve measure. If a set $A$ with $1>\mu(A)>0$ is such that the equation $$\mu(\bigcup_{n\in\mathbb{N}}\,T^{-n}\,A)=1 $$ hold. Is it true that $$\mu(\bigcup_{n\in\mathbb{N}}\,T^{-n}\,X\backslash A)=1 ?$$ That is for the complement of $A$ the equation holds as well.

Answer 1

No! Here is a counter-example:

Let $X:=\{0,1,2\}$ and define $T:X\to X$ with $T(0)=0$, $T(1)=2$ and $T(2)=1$, so that $T$ has a fixed point $0$ and a periodic cycle $1\rightleftarrows 2$. With the discrete topology on $X$, the map $T$ is continuous. It preserves the uniform measure $\mu(\{0\})=\mu(\{1\})=\mu(\{2\})=1/3$. The set $A:=\{0,1\}$ satisfies $\bigcup_{n\in\mathbb{N}}T^{-n}A=X$ but $\bigcup_{n\in\mathbb{N}}T^{-n}(X\setminus A)=\{1,2\}$.