Integral of Survival Function
Jump to navigation
Jump to search
This theorem requires a proof..
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $13.11$
