Difference between revisions of "Notes:Halmos measure theory skeleton"
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*** {{M|1=\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\}]}} then {{M|\mu^*}} is an ''extension'' of {{M|\mu}} to an outer measure on {{M|\mathbf{H}(\mathcal{R})}} | *** {{M|1=\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\}]}} then {{M|\mu^*}} is an ''extension'' of {{M|\mu}} to an outer measure on {{M|\mathbf{H}(\mathcal{R})}} | ||
** {{M|\mu^*}} is the ''outer measure induced by the measure {{M|\mu}}'' | ** {{M|\mu^*}} is the ''outer measure induced by the measure {{M|\mu}}'' | ||
| + | * {{M|\mu^*}}-measurable - given an ''outer measure'' {{M|\mu^*}} on a hereditary {{sigma|ring}} {{M|\mathcal{H} }} a set {{M|A\in\mathcal{H} }} is ''{{M|\mu^*}}-measurable'' if: | ||
| + | ** {{M|1=\forall B\in\mathcal{H}[\mu^*(B)=\mu^*(A\cap B)+\mu^*(B\cap A')]}} | ||
| + | *** '''PROBLEM: How can we do [[complementation]] in a ring?''' | ||
Revision as of 19:26, 22 March 2016
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
- 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]
- Theorem: If [ilmath]\mu[/ilmath] is a measure on a ring [ilmath]\mathcal{R} [/ilmath] and if:
- [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?
- [ilmath]\forall B\in\mathcal{H}[\mu^*(B)=\mu^*(A\cap B)+\mu^*(B\cap A')][/ilmath]
