Non-Negative Signed Measure is Measure

Theorem

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

Let $\mu$ be a signed measure on $\struct {X, \Sigma}$ such that:

$\map \mu A \ge 0$

for each $A \in \Sigma$.

Then $\mu$ is a measure on $\struct {X, \Sigma}$.

Proof

We verify each of the conditions given in the definition of a measure.

From the definition of a signed measure, $\mu$ is a function $\Sigma \to \overline \R$.

Proof of $(1)$

By hypothesis we have:

$\map \mu A \ge 0$

for each $A \in \Sigma$, so condition $(1)$ is satisfied.

$\Box$

Proof of $(2)$

From the definition of a signed measure, we also have that $\mu$ is countably additive, so condition $(2)$ is satisfied.

$\Box$

Proof of $(3')$

Finally, from the definition of a signed measure, we have:

$\map \mu \O = 0$

$\Box$

So $\mu$ is a measure.

$\blacksquare$