Linear Combination of Measures

Theorem
Let $\left({X, \mathcal A}\right)$ be a measurable space.

Let $\mu, \nu$ be measures on $\left({X, \mathcal A}\right)$.

Then for all $a, b \in \R_{\ge 0}$, the positive real numbers:


 * $a \mu + b \nu: \mathcal A \to \overline{\R}, \ \left({a \mu + b \nu}\right) \left({A}\right) = a \mu \left({A}\right) + b \nu \left({A}\right)$

is also a measure on $\left({X, \mathcal A}\right)$.