Revision as of 01:55, 26 July 2015

Definition

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$ A (positive)^{[Note 1]} measure^[1] on a measurable space $(X,\mathcal{A})$ (where recall $X$ is a set and $\mathcal{A}$ is a $\sigma$-algebra on that set) is a mapping:

• $$\mu:\mathcal{A}\rightarrow[0,\infty]$$

That satisfies:

1. $\mu(\emptyset)=0$
2. For any finite sequence of pairwise disjoint sets $(A_i)_{i=1}^n\subseteq\mathcal{A}$ we have $\mu\left(\udot_{i=1}^nA_i\right)=\sum^n_{i=1}\mu(A_i)$
3. For any countably infinite sequence of pairwise disjoint sets $(A_n)_{n=1}^\infty\subseteq\mathcal{A}$ we have $\mu\left(\udot_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\mu(A_n)$

Terminology

TODO: Find references

Of sets

Of measures

Contrast with pre-measure

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

Properties

Here $(X,\mathcal{A},\mu)$ is a measure space, and $A,B\in\mathcal{A}$

Related theorems

• A function is a measure iff it measures the empty set as 0, disjoint sets add, and it is continuous from below (with equiv. conditions)

See also

• Pre-measure

Notes

1. Jump up ↑ What else is there? Measures, Integrals and Martingales mentions this
2. Jump up ↑ Sometimes stated as monotone (it is monotone in Measures, Integrals and Martingales in fact!)

References

1. Jump up ↑ Measures, Integrals and Martingales - Rene L. Schilling

Old page

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. That is:

• $\mu(\emptyset)=0$
• $$\mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n)$$
• $$\mu(S)\ge 0\ \forall S\in\mathcal{A}$$

Contrast with pre-measure

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

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