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 $F_f : \R \to \R$ be the survival function of $f$.

Then:

$\ds \int f \rd \mu = \int_{\openint 0 \to} F_f \rd \lambda$

where $\lambda$ is Lebesgue measure.

Proof

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Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $13.11$