Characterization of Almost Everywhere Zero

Theorem
Let $\struct {X, \Sigma}$ be a measurable space.

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

Let $f : X\to\R$ be a measurable function.

Then:
 * $\forall A\in\Sigma : \int _A f d \mu= 0$
 * $\iff$
 * $f=0$ $\mu$-almost everywhere.

Necessary condition
This is trivial.

Sufficient condition
We show the contraposition.

Assume that $f$ is not zero $\mu$-almost everywhere.

That is:
 * $\map \mu {\set { f > 0 }} > 0$

or:
 * $\map \mu {\set { f < 0 }} > 0$

Therefore:
 * $\int _{\set { f > 0 }} f d \mu > 0$

or:
 * $\int _{\set { f < 0 }} f d \mu < 0$

respectively.