Let $f,g\in\mathbb{L}(E)$. Suppose that $f\leq g$ and $A:=${$x\in E| f(x)<g(x)$}. Prove that $\int_{E}f<\int_{E}g$ if and only if $A$ has positive measure.
2026-05-05 14:43:11.1777992191
Inequality of Lebesgue integrals
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I think you wrote the inequality of the result in the wrong way.
You can write $E$ as $E=A\cup A^c$ where $A^c=\{x\in E: f\geqslant g\}=\{x\in E: f=g\}$ (by hypothesis on $f$ and $g$).
You decompose the integral with these two sets, and then, with a bit more details the result is proven.