Let $X$ be a separable metric space with a Borel regular outer measure $\mu^{*}$ such that $\mu^{*}X = 1 $.
I want to prove that there exists a measure preserving transformation $f : X \to [0;1]$ such that for every Open ball $B$ in $X$, there exists a subball $B^{*}$ such that $\mu(B^{*}) = m(f(B^{*}))$, where $m$ is Lebesgue-measure.
I have really a trouble in constructing such mapping, I think it will be enough too if I just show that such mapping exists, without constructing it.
Any help would be appreciated.