Definition:Series of Measures

Definition
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 $\left({\lambda_n}\right)_{n \in \N}$ be a sequence of positive real numbers.

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


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

is called a series of measures.

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


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

thus implicitly defining the sequences $\left({\mu_n}\right)_{n \in \N}$ and $\left({\lambda_n}\right)_{n \in \N}$.

Examples

 * Discrete Measure

Also see

 * Series of Measures is Measure