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.
How to show that two equimeasurable functions are both integrable or both not integrable
1
$\begingroup$
real-analysis
measure-theory
1 Answers
5
$\int_Xf(x)\mathrm d\mu(x)=\int_0^{+\infty}\mu\{x\in X\,:\,f(x)\geqslant t\}\,\mathrm dt $
-
0Thanks. If I well remember I saw such formula in books from probability theory. – 2011-12-19