given probability space $ ([0,1], \mathcal{B}, \mathbb{P}=Leb)$, a double map $ T=\times 2 (\mod 1): [0,1] \to [0,1]$.
my question is:
can I construct a random variable $ X:[0,1]\to \mathbb{R}$ s.t. $X$ is independent of $T^{-1}\mathcal{B}$
i.e. $\mathbb{P}(X \in \cdot \cap T^{-1}A)=\mathbb{P}(X \in \cdot )\cdot\mathbb{P}(T^{-1}A) $ for any Borel set $A\in \mathcal{B}$?
further question: double map preserves $Leb$, and uniform. but if replace double map with$ f(x)=x+\sqrt{2}x^{\frac{3}{2}} \text{ if } x\in [0, \frac{1}{2}], 2x-1 \text{ if } x\in [\frac{1}{2}, 1]$. Can we find such random variable?
Thank!
Sure. Take $X = \mathbf{1}_{x > 1/2}$. Then for each $A \subseteq [0,1]$, $$ \mathbb P (T \in A) = \mathbb P (x \in \frac 12 A \text{ or } x \in \frac 12 (A+1)) = |A|, $$ while $$ \mathbb P(T \in A, X = 0) = \mathbb P(x \in \frac 12 A) = \frac 12 |A|. $$