Category:Discrete Measure

This category contains results about Discrete Measure.

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

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

That is, if and only if there exist:

a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $X$

and:

a sequence $\sequence {\lambda_n}_{n \mathop \in \N}$ in $\R$

such that:

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

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