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)

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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]

  1. [ilmath]\mu(\emptyset)=0[/ilmath]
  2. [ilmath]\mu(A\udot B)=\mu(A)+\mu(B)[/ilmath]
  3. Either:
    1. 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]
    2. Or [ilmath]\forall A\in\mathcal{A} [/ilmath] we have [ilmath]\mu(A)<\infty[/ilmath] AND:
      1. Either (these are equivalent)[1][Note 2]
        1. 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]
        2. 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]

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

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

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Measures, Integrals and Martingales - Rene L. Schilling



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