Definition:Discrete Measure

Definition

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 $x_n$.

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 $\ds \mu := \sum_{n \mathop \in \N} \lambda_n \delta_{x_n}$ 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

• Results about discrete measures can be found here.

Linguistic Note

Be careful with the word discrete.

A common homophone horror is to use the word discreet instead.

However, discreet means cautious or tactful, and describes somebody who is able to keep silent for political or delicate social reasons.