Difference between revisions of "Measure"

Revision as of 22:48, 30 March 2016

Definition

A (positive) measure, [ilmath]\mu[/ilmath] is a set function from a [ilmath]\sigma[/ilmath]-ring, [ilmath]\mathcal{R} [/ilmath], to the positive extended real values^{[Note 1]}, [ilmath]\bar{\mathbb{R} }_{\ge 0} [/ilmath]^[1][2][3]:

• [ilmath]\mu:\mathcal{R}\rightarrow\bar{\mathbb{R} }_{\ge0} [/ilmath]

Such that:

• [ilmath]\forall(A_n)_{n=1}^\infty\subseteq\mathcal{R}\text{ pairwise disjoint }[\mu\left(\bigudot_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\mu(A_n)][/ilmath] ([ilmath]\mu[/ilmath] is a countably additive set function)
  • Recall that "pairwise disjoint" means [ilmath]\forall i,j\in\mathbb{N}[i\ne j\implies A_i\cap A_j=\emptyset][/ilmath]

Entirely in words a (positive) measure, [ilmath]\mu[/ilmath] is:

• An extended real valued countably additive set function from a [ilmath]\sigma[/ilmath]-ring, [ilmath]\mathcal{R} [/ilmath]; [ilmath]\mu:\mathcal{R}\rightarrow\bar{\mathbb{R} } [/ilmath].

Remember that every [ilmath]\sigma[/ilmath]-algebra is a [ilmath]\sigma[/ilmath]-ring, so this definition can be applied directly (and should be in the reader's mind) to [ilmath]\sigma[/ilmath]-algebras

Terminology

For a set

We may say a set [ilmath]A\in\mathcal{R} [/ilmath] (for a [ilmath]\sigma[/ilmath]-ring [ilmath]\mathcal{R} [/ilmath]) is:

Of a measure

We may say a measure, [ilmath]\mu[/ilmath] is:

Of a measure on a [ilmath]\sigma[/ilmath]-algebra

If [ilmath]\mu:\mathcal{A}\rightarrow\bar{\mathbb{R} }_{\ge0} [/ilmath] for a [ilmath]\sigma[/ilmath]-algebra [ilmath]\mathcal{A} [/ilmath]^{[Note 2]} then we can define:

Immediate properties

Properties

TODO: Countable subadditivity and so forth

In common with a pre-measure

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)

Examples

• Dirac measure
• Counting measure
• Discrete probability measure
• Lebesgue measure

Trivial measures

Here [ilmath]\mathcal{R} [/ilmath] is a [ilmath]\sigma[/ilmath]-ring^{[Note 4]}

1. [math]\mu:\mathcal{R}\rightarrow\{0,+\infty\}[/math] by [math]\mu(A)=\left\{\begin{array}{lr} 0 & \text{if }A=\emptyset \\ +\infty & \text{otherwise} \end{array}\right.[/math]
   • Note that if we'd chosen a finite and non-zero value instead of [ilmath]+\infty[/ilmath] it would not be a measure^{[Note 5]}, as take any non-empty [ilmath]A,B\in\mathcal{R} [/ilmath] with [ilmath]A\cap B=\emptyset[/ilmath], for a measure we would have:
     • [ilmath]\mu(A\cup B)=\mu(A)+\mu(B)[/ilmath], which will yield [ilmath]v=2v\implies v=0[/ilmath] contradicting that [ilmath]\mu[/ilmath] maps non-empty sets to finite non-zero values
2. [math]\mu:\mathcal{R}\rightarrow\{0\}[/math] by [math]\mu:A\mapsto 0[/math] is the trivial measure.

See also

• Pre-measure
• Outer-measure
• Constructing a measure from a pre-measure
• Measurable space
• Measure space

Notes

1. ↑ Recall [ilmath]\bar{\mathbb{R} }_{\ge0} [/ilmath] is [ilmath]\mathbb{R}_{\ge0}\cup\{+\infty\} [/ilmath]
2. ↑ Remember a sigma-algebra is just a sigma-ring containing the entire space.
3. ↑ Sometimes stated as monotone (it is monotone in Measures, Integrals and Martingales in fact!)
4. ↑ Remember every [ilmath]\sigma[/ilmath]-algebra is a [ilmath]\sigma[/ilmath]-ring, so [ilmath]\mathcal{R} [/ilmath] could just as well be a [ilmath]\sigma[/ilmath]-algebra
5. ↑ Unless [ilmath]\mathcal{R} [/ilmath] was a trivial [ilmath]\sigma[/ilmath]-algebra consisting of the empty set and another set.

References

Note: Inline with the Measure theory terminology doctrine the references do not define a measure exactly as such, only an object that fits the place we have named measure. This sounds like a huge discrepancy but as is detailed on that page, it isn't.

1. ↑ ^{1.0 1.1 1.2 1.3 1.4 1.5 1.6} Measure Theory - Paul R. Halmos
2. ↑ Measures, Integrals and Martingales - René L. Schilling
3. ↑ Measure Theory - Volume 1 - V. I. Bogachev