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

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