Distribution function of a measurable function

Summary

The distribution function of a measurable function $f$ records, for each level $t \ge 0$, the measure of the set where $\abs{f}$ is larger than $t$. It forgets where the large values of $f$ sit and keeps only how much room they take. Many quantities, such as the $L^p$ norms, depend on $f$ only through its distribution function. It is the starting point for the layer cake formula, rearrangements, weak $L^p$ spaces and interpolation.