Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and

- g is radially symmetric,
- the function $ (0, \infty )\ni t \mapsto g (tx)$ is increasing and
- their distribution functions coincide, i.e., for all values $\lambda >0$ : $\vert \{ f(y) < \lambda \} \vert = \vert \{ g(y) < \lambda \} \vert < \infty$, where $\vert A \vert$ denotes the Lebesgue measure of $A \subset \mathbb{R}^n$.

I would like to show that there is a measure-preserving bijection $\phi$ such that $ g = f \circ \phi$.

So far, I have no idea how to tackle this problem. I would be really grateful for any suggestions. Thanks!