# 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)

## Contents

## Statement

[math]\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}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]Let [ilmath](X,\mathcal{A})[/ilmath] be a measurable space. A map:

- [ilmath]\mu:\mathcal{A}\rightarrow[0,\infty][/ilmath]

is a measure *if and only if*^{[1]}

- [ilmath]\mu(\emptyset)=0[/ilmath]
- [ilmath]\mu(A\udot B)=\mu(A)+\mu(B)[/ilmath]
- Either:
- For any increasing sequence of sets
^{[Note 1]}[ilmath](A_n)_{n=1}^\infty\subseteq\mathcal{A}[/ilmath] with [ilmath]\lim_{n\rightarrow\infty}(A_n)=A\in\mathcal{A}[/ilmath] we have- [math]\mu(A)=\lim_{n\rightarrow\infty}(\mu(A_n))=\inf_{n\in\mathbb{N} }(\mu(A_n))[/math]
- This is called
*Continuity of measures from below*^{[1]}

- Or [ilmath]\forall A\in\mathcal{A} [/ilmath] we have [ilmath]\mu(A)<\infty[/ilmath] AND:
- Either (these are equivalent)
^{[1]}^{[Note 2]}- For any decreasing sequence of sets
^{[Note 3]}[ilmath](A_n)_{n=1}^\infty\subseteq\mathcal{A}[/ilmath] with [ilmath]\lim_{n\rightarrow\infty}(A_n)=A\in\mathcal{A}[/ilmath] we have- [math]\mu(A)=\lim_{n\rightarrow\infty}(\mu(A_n))=\inf_{n\in\mathbb{N} }(\mu(A_n))[/math]
- This is called
*Continuity of measures from above*^{[1]}

- For any decreasing sequence of sets [ilmath](A_n)_{n=1}^\infty[/ilmath] with [ilmath]\lim_{n\rightarrow\infty}(A_n)=\emptyset[/ilmath] we have:
- [math]\lim_{n\rightarrow\infty}(\mu(A_n))=0[/math]
- This is called
*continuity of measures at [ilmath]\emptyset[/ilmath]*^{[1]}

- For any decreasing sequence of sets

- Either (these are equivalent)

- For any increasing sequence of sets

## Page notes

This is actually several theorems rolled into one. Halmos has some good terminology and splits these theorems up. I will come back to this when I've done that.

As it stands now this is a *good* theorem with some extra facts bolted on.
I like conditions 1 2 and 3.1 [ilmath]\iff[/ilmath] [ilmath]\mu[/ilmath] is a measure.

## Proof

From^{[1]} page 24 - although not hard to do without.

TODO: Clean up and prove

## Notes

- ↑ A sequence of sets [ilmath](A_n)_{n=1}^\infty[/ilmath] is increasing if [ilmath]A_n\subseteq A_{n+1} [/ilmath]
- ↑ Check/prove this
- ↑ A sequence of sets [ilmath](A_n)_{n=1}^\infty[/ilmath] is decreasing if [ilmath]A_{n+1}\subseteq A_n[/ilmath]

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}Measures, Integrals and Martingales - Rene L. Schilling