Measure

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$ Not to be confused with Pre-measure

Definition

A $\sigma$-ring $\mathcal{A}$ and a countably additive, extended real valued. non-negative set function $$\mu:\mathcal{A}\rightarrow[0,\infty]$$ is a measure.

Contrast with pre-measure

Note: the family $$A_n$$ must be pairwise disjoint

Property	Measure	Pre-measure
	$$\mu:\mathcal{A}\rightarrow[0,\infty]$$	$$\mu_0:R\rightarrow[0,\infty]$$
	$$\mu(\emptyset)=0$$	$$\mu_0(\emptyset)=0$$
Finitely additive	$$\mu\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu(A_i)$$	$$\mu_0\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu_0(A_i)$$
Countably additive	$$\mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n)$$	If $$\bigudot^\infty_{n=1}A_n\in R$$ then $$\mu_0\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu_0(A_n)$$

Terminology

These terms apply to pre-measures to, rather $\mathcal{A}$ you would use the ring the pre-measure is defined on.

Complete measure

A measure is complete if for $A\in\mathcal{A}$ we have $$[\mu(A)=0\wedge B\subset A]\implies B\in \mathcal{A}$$

Finite

A set $A\in\mathcal{A}$ is finite if $\mu(A)<\infty$ - we say "$A$ has finite measure"

Finite measure

$\mu$ is a finite measure if every set $\in\mathcal{A}$ is finite.

Sigma-finite

A set $A\in\mathcal{A}$ is $\sigma$-finite if $$\exists(A_n)_{n=1}^\infty:[A\subseteq\cup^\infty_{n=1}A_n\wedge(\forall A_n,\ \mu(A_n)<\infty)]$$

Sigma-finite measure

$\mu$ is $\sigma$-finite if every set $\in\mathcal{A}$ is $\sigma$-finite

Total

If $\mathcal{A}$ is a $\sigma$-algebra rather than a ring (that is $X\in\mathcal{A}$ where $X$ is the space) then we use

Totally finite measure

If $X$ is finite

Totally sigma-finite measure

If $X$ is $\sigma$-finite

Examples

• Dirac measure
• Counting measure
• Discrete probability measure

Trivial measures

Given the Measurable space $(X,\mathcal{A})$ we can define:

$$\mu:\mathcal{A}\rightarrow\{0,+\infty\}$$ by $$\mu(A)=\left\{\begin{array}{lr} 0 & \text{if }A=\emptyset \\ +\infty & \text{otherwise} \end{array}\right.$$

Another trivial measure is:

$$v:\mathcal{A}\rightarrow\{0\}$$ by $$v(A)=0$$ for all $$A\in\mathcal{A}$$

See also

• Pre-measure
• Outer-measure
• Lebesgue measure