View source for Notes:Halmos measure theory skeleton

==Skeleton==
* [[Ring of sets]]
* [[Sigma-ring]]
* [[additive set function]]
* ''measure'', {{M|\mu}} - [[extended real valued]], non negative, [[countably additive set function]] defined on a [[ring of sets]]
* ''hereditary'' system - a system of sets, {{M|\mathcal{E} }} such that if {{M|E\in\mathcal{E} }} then {{M|\forall F\in\mathcal{P}(E)[F\in\mathcal{E}]}}
** ''hereditary ring generated by''
* ''subadditivity''
* ''outer measure'', {{M|\mu^*}} (p42) - [[extended real valued]], non-negative, monotone and countably subadditive set function on an ''hereditary'' {{sigma|ring}} with {{M|1=\mu^*(\emptyset)=0}}
** Theorem: If {{M|\mu}} is a measure on a ring {{M|\mathcal{R} }} and if:
*** {{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^*}}-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?'''