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

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
 * $\displaystyle \int \size f \rd \mu = 0$