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:
 * $\ds \forall A \in \Sigma : \int_A f \rd \mu = 0$


 * $f = 0$ $\mu$-almost everywhere.
 * $f = 0$ $\mu$-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:
 * $\ds \int_{\set {f \mathop > 0} } f \rd \mu > 0$

or:
 * $\ds \int_{\set {f \mathop < 0} } f \rd \mu < 0$

respectively.