Integral of Survival Function
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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.
Proof
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $13.11$