Distribution function of a measurable function

Remarks

Example: let $X = [0, 1]$ with Lebesgue measure and $f(x) = x^2$. For $0 \le t < 1$ the set $\set{x : x^2 > t}$ is $(\sqrt{t}, 1]$, so $d_f(t) = 1 - \sqrt{t}$. For $t \ge 1$ the set is empty, so $d_f(t) = 0$.

Non-example: the cumulative distribution function $F(t) = \mu(\set{f \le t})$ of probability theory is a different object. It measures where $f$ is small, uses $f$ and not $\abs{f}$, and is increasing rather than decreasing.

The function $d_f$ is non-increasing and right-continuous. Two functions with the same distribution function are called equimeasurable, and they have the same $L^p$ norms. Some authors write $\lambda_f$ or $\mu_f$ for $d_f$.