Characterization of Almost Everywhere Zero

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

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

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

Then:
 * $\ds \forall A \in \Sigma : \int \paren {\chi_A \cdot f} \rd \mu = 0$


 * $f = 0$ $\mu$-almost everywhere
 * $f = 0$ $\mu$-almost everywhere

where:
 * $\chi_A$ is the characteristic function of $A$

Necessary condition
This is trivial.

Sufficient condition
Similarly:
 * $\map\mu{\set {-f > \frac{1}{n} } } = 0$

that implies:
 * $\map\mu{\set {\size f > \frac{1}{n} } } = 0$

Therefore: