Definition:Series of Measures

Definition

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

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

Let $\sequence {\lambda_n}_{n \mathop \in \N}$ be a sequence of positive real numbers.

Then the mapping $\mu: \Sigma \to \overline \R$, defined by:

$\ds \map \mu E := \sum_{n \mathop \in \N} \lambda_n \map {\mu_n} E$

is called a series of measures.

When introducing a series of measures, it is convenient and common to do this by a phrase of the form:

'Let $\mu := \ds \sum_{n \mathop \in \N} \lambda_n \mu_n$ be a series of measures.'

thus implicitly defining the sequences $\sequence {\mu_n}_{n \mathop \in \N}$ and $\sequence {\lambda_n}_{n \mathop \in \N}$.