If $x \in(0,1]$, set

$$x=\frac{1}{N(x)+T(x)}$$

where $N(x)$ is an integer and $1>T(x) \geqslant 0$. Let $N(0)=T(0)=0$.

If $x$ is also irrational, write down the continued fraction expansion in terms of $N T^{j}(x)\left(\right.$ where $\left.N T^{0}(x)=N(x)\right)$.

Let $X$ be a random variable taking values in $[0,1]$ with probability density function

$$f(x)=\frac{1}{(\log 2)(1+x)}$$

Show that $T(X)$ has the same distribution as $X$.