Notes:Halmos measure theory skeleton/Old page

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Halmos' measure theory: skeleton

Skeleton

• Ring of sets
• Sigma-ring
• additive set function
• measure, [ilmath]\mu[/ilmath] - extended real valued, non negative, countably additive set function defined on a ring of sets
  • complete measure - [ilmath]\mu[/ilmath] is complete if:
    • [ilmath]\forall A\in\mathcal{R}\forall B[B\subseteq A\wedge\mu(A)=0\implies B\in\mathcal{R} ][/ilmath]
• hereditary system - a system of sets, [ilmath]\mathcal{E} [/ilmath] such that if [ilmath]E\in\mathcal{E} [/ilmath] then [ilmath]\forall F\in\mathcal{P}(E)[F\in\mathcal{E}][/ilmath]
  • hereditary ring generated by
• subadditivity
• outer measure, [ilmath]\mu^*[/ilmath] (p42) - extended real valued, non-negative, monotone and countably subadditive set function on an hereditary [ilmath]\sigma[/ilmath]-ring with [ilmath]\mu^*(\emptyset)=0[/ilmath]
  • Theorem: If [ilmath]\mu[/ilmath] is a measure on a ring [ilmath]\mathcal{R} [/ilmath] and if:
    • [ilmath]\forall A\in\mathbf{H}(\mathcal{R})[\mu^*(A)=\text{inf}\{\sum^\infty_{n=1}\mu(A_n)\ \vert\ (A_n)_{n=1}^\infty\subseteq\mathcal{R} \wedge A\subseteq \bigcup^\infty_{n=1}A_n\}][/ilmath] then [ilmath]\mu^*[/ilmath] is an extension of [ilmath]\mu[/ilmath] to an outer measure on [ilmath]\mathbf{H}(\mathcal{R})[/ilmath]
  • [ilmath]\mu^*[/ilmath] is the outer measure induced by the measure [ilmath]\mu[/ilmath]
• [ilmath]\mu^*[/ilmath]-measurable - given an outer measure [ilmath]\mu^*[/ilmath] on a hereditary [ilmath]\sigma[/ilmath]-ring [ilmath]\mathcal{H} [/ilmath] a set [ilmath]A\in\mathcal{H} [/ilmath] is [ilmath]\mu^*[/ilmath]-measurable if:
  • [ilmath]\forall B\in\mathcal{H}[\mu^*(B)=\mu^*(A\cap B)+\mu^*(B\cap A')][/ilmath]
    • PROBLEM: How can we do complementation in a ring?
    • Solution: Note that [ilmath]S\cap T'=S-T[/ilmath], so Halmos is really saying:
      • [ilmath]A\in\mathcal{H}[/ilmath] is [ilmath]\mu^*[/ilmath]-measurable if:
        • [ilmath]\forall B\in\mathcal{H}[\mu^*(B)=\mu^*(A\cap B)+\mu^*(B - A)][/ilmath]
• Theorem: if [ilmath]\mu^*[/ilmath] is an outer measure on a hereditary [ilmath]\sigma[/ilmath]-ring [ilmath]\mathcal{H} [/ilmath] and if [ilmath]\mathcal{S} [/ilmath] is the class of all [ilmath]\mu^*[/ilmath]-measurable sets, then [ilmath]\mathcal{S} [/ilmath] is a ring of sets
• Theorem: (p46) - [ilmath]\mathcal{S} [/ilmath] is a sigma-ring
• Theorem: - Every set of [ilmath]\mu^*=0[/ilmath] belongs to [ilmath]\mathcal{S} [/ilmath] and the set function [ilmath]\bar{\mu}:\mathcal{S}\rightarrow\bar{\mathbb{R} }_{\ge 0} [/ilmath] is a complete measure (AKA: [ilmath]\bar{\mu} [/ilmath] is the measure induced by [ilmath]\mu^*[/ilmath]).
• Theorem - Every set in [ilmath]\sigma_R(\mathcal{R})[/ilmath] is [ilmath]\mu^*[/ilmath]-measurable (the [ilmath]\sigma[/ilmath]-ring generated by [ilmath]\mathcal{R} [/ilmath])
  • Alternatively: [ilmath]\sigma_R(\mathcal{R})\subseteq\mathcal{S} [/ilmath]
• Theorem - [ilmath]\forall A\in\mathbf{H}(\mathcal{R})[\underbrace{\mu^*(A)}_{\text{Outer} }=\underbrace{\text{inf}\{\bar{\mu}(B)\ \vert\ A\subseteq B\in\mathcal{S}\} }_{\text{Induced by } \bar{\mu} }=\underbrace{\{\bar{\mu}(B)\ \vert\ A\subseteq B\in\sigma_R(\mathcal{R})\} }_{\text{Induced by } \bar{\mu} }][/ilmath]
  • That is to say that:
    • Outer measure induced by [ilmath]\bar{\mu} [/ilmath] on [ilmath]\sigma_R(\mathcal{R})[/ilmath] AND
    • the outer measure induced by [ilmath]\bar{\mu} [/ilmath] on [ilmath]\mathcal{S} [/ilmath]
    • agree with [ilmath]\mu^*[/ilmath]
• completion of a measure - p55
• Theorem: if [ilmath]\mu[/ilmath] is a [ilmath]\sigma[/ilmath]-finite measure on a [ilmath]\sigma[/ilmath]-ring [ilmath]\mathbb{R} [/ilmath] and if [ilmath]\mu^*[/ilmath] is the outer measure it induces, then:
  • The completion of the extension of [ilmath]\mu[/ilmath] to [ilmath]\sigma_R(\mathcal{R})[/ilmath] is identical with [ilmath]\mu^*[/ilmath] on the class of all [ilmath]\mu^*[/ilmath]-measurable sets