Measure

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 $\Sigma$,

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.