Integrable Function Zero A.E. iff Absolute Value has Zero Integral

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Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $f: X \to \overline{\R}$ be a $\mu$-integrable function.


Then the following are equivalent:

$f = 0$ almost everywhere
$\displaystyle \int \left\vert{f}\right\vert \, \mathrm d \mu = 0$


Proof


Sources