Discrete measure

Definition and properties

Given two (positive) σ-finite measures ${\displaystyle \mu }$ and ${\displaystyle \nu }$ on a measurable space ${\displaystyle (X,\mathrm{\Sigma })}$. Then ${\displaystyle \mu }$ is said to be discrete with respect to ${\displaystyle \nu }$ if there exists an at most countable subset ${\displaystyle S\subset X}$ in ${\displaystyle \mathrm{\Sigma }}$ such that

1. All singletons ${\displaystyle \{s\}}$ with ${\displaystyle s\in S}$ are measurable (which implies that any subset of ${\displaystyle S}$ is measurable)
2. ${\displaystyle \nu (S)=0}$
3. ${\displaystyle \mu (X\setminus S)=0.}$

A measure ${\displaystyle \mu }$ on ${\displaystyle (X,\mathrm{\Sigma })}$ is discrete (with respect to ${\displaystyle \nu }$) if and only if ${\displaystyle \mu }$ has the form

${\displaystyle \mu =\sum _{i=1}^{\mathrm{\infty }}{a}_i{\delta }_{s_i}}$

with ${\displaystyle a_i\in {\mathbb{R}}_{>0}}$ and Dirac measures ${\displaystyle {\delta }_{s_i}}$ on the set ${\displaystyle S=\{s_i{\}}_{i\in \mathbb{N}}}$ defined as

${\displaystyle {\delta }_{s_i}(X)=\{\begin{array}{ll}1& \text{ if }{s}_i\in X\\ 0& \text{ if }{s}_i\notin X\end{array}}$

for all ${\displaystyle i\in \mathbb{N}}$.

One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that ${\displaystyle \nu }$ be zero on all measurable subsets of ${\displaystyle S}$ and ${\displaystyle \mu }$ be zero on measurable subsets of ${\displaystyle X\mathrm{\setminus }S.}$