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

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Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

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


Then the following are equivalent:

$f = 0$ almost everywhere
$\displaystyle \int \size f \rd \mu = 0$


Proof


Sources