Let $f \, \colon \mathbb{R}^d \rightarrow[0, \infty)$ be a function such that the sets $ \{ y \: \colon f(y) > \lambda \}$ are of finite Lebesgue-measure, for every $\lambda \geq 0$. Then, we can define the spherically symmetric and decreasing rearrangement $f^* \, \colon \mathbb{R}^d \rightarrow \mathbb{R}$ of $f$ by \begin{equation*} f^* (x) = \int_{0}^{\infty} \chi_{ \{f > t \}^*} (x) \, \textrm{d}t. \end{equation*} Due to a theorem of John V. Ryff, in the $1$-dimensional case, there is a measure-preserving map $\sigma$ such that \begin{equation*} f= f^* \circ \sigma \quad \textrm{ a.e.} \end{equation*} (In fact, this is a corollary of the cited theorem and can be found in [1, Corollary 7.6 in chapter 2].) My question is now:
Is there a similar statement when $d \geq 2$?
I expect that the answer is yes, but, unfortunately, I haven't found a reference. I would be grateful for any suggestions. Thanks!
[1] Bennett, C., Sharpley, R. Interpolation of operators
