Definition:Discrete Measure

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

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

That is, iff there exist:


 * A sequence $\left({x_n}\right)_{n \in \N}$ in $X$
 * A sequence $\left({\lambda_n}\right)_{n \in \N}$ in $\R$

such that:


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

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 $\left({x_n}\right)_{n \in \N}$ and $\left({\lambda_n}\right)_{n \in \N}$.

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

Also see

 * Series of Measures