Suppose $f$ is a measurable function over $[0,1]$ (for simplicity sake). Let $f^*$ be the associated decreasing rearrangement function. We can determine $f$ using a measure preserving map $\phi:[0,1] \rightarrow [0,1]$ defined in this paper as follows:
$$ \phi(t) = \mu ( \{x: f(x) > f(t) \}) + \mu ( x: f(x) = f(t), \ x \leq t ). $$
It turns out that $f^* \circ \phi = f$. Unfortunately, It is not always the case that $\phi$ is bijective or bimeasurable, i.e., $\phi$ is not a measure preserving transformation. However, under what conditions is $\phi$ a transformation?
For instance, it seems to be the case when $f$ is a simple function spanned by characteristic functions of intervals. But what about other such cases?
