Integral of Increasing Function Composed with Measurable Function

Theorem
Let $\struct {X, \Sigma, \mu}$ 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 $\map \phi 0 = 0$.

Then:


 * $\ds \int \phi \circ f \rd \mu = \int_0^\infty \map {\phi'} t \map F t \rd t$

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