Measure - Knowlpedia

A measure on a measurable space $(X,\Sigma)$ is a function $\mu:\Sigma\to[0,\infty]$ such that $\mu(\varnothing)=0$ and for every pairwise disjoint sequence $(A_n)_{n\ge 1}$ in $\Sigma$ ,

\mu\!\left(\bigcup_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty \mu(A_n).

A measure assigns “sizes” to measurable sets and turns $(X,\Sigma)$ into a measure space. Sets of measure zero are null sets, and they control notions like almost everywhere.

Examples

The counting measure on $(X,\mathcal P(X))$ is given by $\mu(A)=|A|$ (possibly $+\infty$ ) for any subset $A\subseteq X$ .
Lebesgue measure on $\mathbb R^n$ is the standard measure extending ordinary length/area/volume.
For a point $x_0\in X$ , the Dirac measure $\delta_{x_0}$ is defined by $\delta_{x_0}(A)=1$ if $x_0\in A$ and $0$ otherwise.