Series of Measures is Measure

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

Let $\left({\mu_n}\right)_{n \in \N}$ be a sequence of measures on $\left({X, \Sigma}\right)$.

Let $\left({a_n}\right)_{n \in \N} \subseteq \R_{\ge 0}$ be a sequence of positive real numbers.

Then the series of measures $\mu: \Sigma \to \overline{\R}$, defined by:


 * $\displaystyle \mu \left({E}\right) := \sum_{n \mathop \in \N} a_n \mu_n \left({E}\right)$

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