Integral of Survival Function

Theorem

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

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

Let $F \left({f}\right): \R \to \R$ be the survival function of $f$.

Then:

$\displaystyle \int f \, \mathrm d \mu = \int_{\left({0 \,.\,.\, \infty}\right)} F \left({f}\right) \, \mathrm d \lambda$

where $\lambda$ is Lebesgue measure.