Integral of Survival Function

Theorem
Let $\struct {X, \Sigma, \mu}$ be a $\sigma$-finite measure space.

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

Let $\map F f: \R \to \R$ be the survival function of $f$.

Then:


 * $\ds \int f \rd \mu = \int_{\openint 0 \to} \map F f \rd \lambda$

where $\lambda$ is Lebesgue measure.