ARQMATH LAB
Math Stack Exchange

How to show that two equimeasurable functions are both integrable or both not integrable

1
$\begingroup$

Assume that $\mu$ is a positive measure on a $\sigma$-field $S$ of subsets of $X$. Assume that functions $f,g\colon X \to \mathbb{R_+}$ are measurable and satisfy for every $a \in \mathbb{R}$ the following condition: $ \mu \{x\in X: f(x) I would like to ask how to show that either $f,g$ are both integrable and $\int_X f d \mu=\int_X g d\mu$ or $f,g$ are both not integrable.

real-analysis measure-theory
asked 2011-12-19
user id:20924
586
99silver badges 2626bronze badges

1 Answers 1

5

$\int_Xf(x)\mathrm d\mu(x)=\int_0^{+\infty}\mu\{x\in X\,:\,f(x)\geqslant t\}\,\mathrm dt $

asked 2011-12-19
user id:6179
250k
2323gold badges 219219silver badges 453453bronze badges
  • 0
    Thanks. If I well remember I saw such formula in books from probability theory. – 2011-12-19

Related Posts

No Related Post Found