Integral of Increasing Function Composed with Measurable Function

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a $\sigma$-finite measure space.

Let $f: X \to \R_{\ge 0}$ be a positive measurable function.

Let $\phi: \R_{\ge 0} \to \R_{\ge 0}$ be a continuously differentiable, increasing function such that $\phi \left({0}\right) = 0$.

Then:


 * $\displaystyle \int \phi \circ f \, \mathrm d \mu = \int_0^\infty \phi' \left({t}\right) F \left({t}\right) \, \mathrm d t$

where:
 * $F$ is the survival function of $f$
 * $\displaystyle \int_0^\infty$ denotes an improper integral.