If $f\in \mathcal{L}_+^0$ and $\int fd\mu=0$, where $\mathcal{L}_+^0$ is the set of all measurable $[0,\infty]$-valued functions, Does that Implies $f=0$ almost everywhere? If so why?
2026-03-25 18:48:06.1774464486
Why does $f\in \mathcal{L}_+^0$ and $\int fd\mu=0\implies f=0$ almost everywhere?
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Hint: \begin{align} \int_{|f|\geq 1/n} f\ d\mu \geq \frac{1}{n}\mu\{x \mid |f|\geq n^{-1}\} \end{align}