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