Series of Measures is Measure

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Theorem

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

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

Let $\left({a_n}\right)_{n \mathop \in \N} \subseteq \R_{\ge 0}$ 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 $\left({X, \Sigma}\right)$.


Proof

Let us verify the conditions for a measure in turn.


Let $E \in \Sigma$.

Then for all $n \in \N$, $\mu_n \left({E}\right) \ge 0$ and so $a_n \mu_n \left({E}\right) \ge 0$.

Therefore, by Series of Positive Real Numbers has Positive Limit:

$\displaystyle \mu \left({E}\right) = \sum_{n \mathop \in \N} a_n \mu_n \left({E}\right) \ge 0$


For every $n \in \N$, also $\mu_n \left({\varnothing}\right) = 0$.

Therefore, it immediately follows that:

$\displaystyle \mu \left({\varnothing}\right) = \sum_{n \mathop \in \N} a_n \mu_n \left({\varnothing}\right) = \sum_{n \mathop \in \N} 0 = 0$


Finally, let $\left({E_n}\right)_{n \mathop \in \N}$ be a sequence of pairwise disjoint sets in $\Sigma$.

Then:

\(\displaystyle \mu \left({\bigcup_{m \mathop \in \N} E_m}\right)\) \(=\) \(\displaystyle \sum_{n \mathop \in \N} a_n \mu_n \left({\bigcup_{m \mathop \in \N} E_m}\right)\) Definition of $\mu$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop \in \N} a_n \left({\sum_{m \mathop \in \N} \mu_n \left({E_m}\right)}\right)\) The $\mu_n$ are measures
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop \in \N} \sum_{m \mathop \in \N} a_n \mu_n \left({E_m}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{m \mathop \in \N} \sum_{n \mathop \in \N} a_n \mu_n \left({E_m}\right)\) Double Series of Positive Real Numbers
\(\displaystyle \) \(=\) \(\displaystyle \sum_{m \mathop \in \N} \mu \left({E_m}\right)\) Definition of $\mu$


Therefore, having verified all three axioms, $\mu$ is a measure.

$\blacksquare$


Sources