Decreasing rearrangement

Summary

The decreasing rearrangement of a measurable function $f$ is the non-increasing, right-continuous function $f^*$ on $[0, \infty)$ that has the same distribution function as $\abs{f}$. It sorts the values of $\abs{f}$ from the largest to the smallest. Every norm that depends only on the distribution function, such as the $L^p$ norms, takes the same value on $f$ and on $f^*$. It is the basic tool of Lorentz spaces and of rearrangement-invariant function spaces.