Definition:Discrete Measure

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Then $\mu$ is said to be a discrete measure it is a series of Dirac measures.

That is, there exist:


 * A sequence $\sequence {x_n}_{n \mathop \in \N}$ in $X$
 * A sequence $\sequence {\lambda_n}_{n \mathop \in \N}$ in $\R$

such that:


 * $(1):\quad \forall E \in \Sigma: \map \mu E = \displaystyle \sum_{n \mathop \in \N} \lambda_n \, \map {\delta_{x_n} } E$

where $\delta_{x_n}$ denotes the Dirac measure at $x_n$.

By Series of Measures is Measure, defining $\mu$ by $(1)$ yields a measure.

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


 * Let $\displaystyle \mu := \sum_{n \mathop \in \N} \lambda_n \delta_{x_n}$ be a discrete measure.

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

Sometimes it is convenient to impose that the sequence $\sequence {x_n}_{n \mathop \in \N}$ is a sequence of distinct terms, that is, that $x_n = x_m$ implies $n = m$.

Also see

 * Series of Measures