Problem on measure theory

Question

Let $ \mu $ be a $ \sigma $-finite measure space on $(X,s)$. Suppose $ f: X \to [0,\infty]$ be a $ s $-measurable and $ p \in [0,\infty]$.

Show that $$ \int_X f^p \, d\mu = \int_{0}^{\infty} pt^{p-1} \mu({x : f(x) > t}) \, dt $$

This is my homework problem. I have no idea how to do this problem..

Have you seen Tonelli's theorem yet? – Umberto P. Apr 27 '15 at 14:06 — Umberto P., Apr 27 '15 at 14:06
@umberto yes,I have seen Tonelli's theorem. – Saikat Basu Apr 27 '15 at 14:07 — Saikat Basu, Apr 27 '15 at 14:07
https://math.stackexchange.com/q/182019/321264 – StubbornAtom Nov 04 '21 at 20:57 — StubbornAtom, Nov 04 '21 at 20:57

score 4 · Answer 1 · answered Apr 27 '15 at 14:08

4

Here is a hint to get started: write $$f^p = \int_0^{f} pt^{p-1} \, dt = \int_0^\infty \chi_{\{t \le f\}} pt^{p-1} \, dt$$ and apply Tonelli's theorem.

answered Apr 27 '15 at 14:08

Umberto P.

54,204

Problem on measure theory

1 Answers1

Linked