Let $X=[0,1]\backslash\mathbb{Q}$ and define the Gauss map $T:X\rightarrow X$ by \begin{equation*} T(x) = \dfrac{1}{x}-\left\lfloor\dfrac{1}{x}\right\rfloor \end{equation*} where $\left\lfloor t\right\rfloor$ denotes the greatest integer less than or equal to $t$. Prove that Gauss map is strong-mixing with respect to the Gauss measure $\mu(A)=\dfrac{1}{\text {log}2} {\displaystyle \int_A }\dfrac{1}{1+x}dx$.
Let $T$ be a measure-preserving tranformation of a probability space $(X,\mathscr{B},\mu)$. We say that $T$ is strong-mixing if $\forall A,B \in $ $\lim_{n\to\infty} \mu(T^{-n}A\cap B)=\mu(A)\mu(B).$