Linear Combination of Signed Measures is Signed Measure

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

Let $\mu$ and $\nu$ be signed measures on $\struct {X, \Sigma}$.

Let $\alpha, \beta \in \overline \R$.

Suppose that the sum:


 * $\alpha \map \mu A + \beta \map \nu A$

is well-defined for each $A \in \Sigma$.

Then:


 * $\xi = \alpha \mu + \beta \nu$ is a signed measure.

Proof
We verify both of the conditions for a signed measure.

Proof of $(1)$
We have:

verifying $(1)$ for $\xi$.

Proof of $(2)$
Let $\sequence {D_n}_{n \mathop \in \N}$ be a sequence of pairwise disjoint $\Sigma$-measurable sets.

Then:

verifying $(2)$ for $\xi$.

So $\xi$ is a signed measure.