Difference between revisions of "Measure"

Latest revision as of 14:39, 16 August 2016

Definition

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

• $\mu:\mathcal{R}\rightarrow\bar{\mathbb{R} }_{\ge0}$

Such that:

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

Entirely in words a (positive) measure, $\mu$ is:

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

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

Terminology

For a set

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

Of a measure

We may say a measure, $\mu$ is:

Of a measure on a $\sigma$-algebra

If $\mu:\mathcal{A}\rightarrow\bar{\mathbb{R} }_{\ge0}$ for a $\sigma$-algebra $\mathcal{A}$^{[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 $\mathcal{R}$ is a $\sigma$-ring^{[Note 4]}

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

See also

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

Notes

1. Jump up ↑ Recall $\bar{\mathbb{R} }_{\ge0}$ is $\mathbb{R}_{\ge0}\cup\{+\infty\}$
2. Jump up ↑ Remember a sigma-algebra is just a sigma-ring containing the entire space.
3. Jump up ↑ Sometimes stated as monotone (it is monotone in Measures, Integrals and Martingales in fact!)
4. Jump up ↑ Remember every $\sigma$-algebra is a $\sigma$-ring, so $\mathcal{R}$ could just as well be a $\sigma$-algebra
5. Jump up ↑ Unless $\mathcal{R}$ was a trivial $\sigma$-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. ↑ ^{Jump up to: 1.0 1.1 1.2 1.3 1.4 1.5 1.6} Measure Theory - Paul R. Halmos
2. Jump up ↑ Measures, Integrals and Martingales - René L. Schilling
3. Jump up ↑ Measure Theory - Volume 1 - V. I. Bogachev