Difference between revisions of "Notes:Halmos measure theory skeleton"

Revision as of 19:26, 22 March 2016

Ring of sets
Sigma-ring
additive set function
measure, $\mu$ - extended real valued, non negative, countably additive set function defined on a ring of sets
hereditary system - a system of sets, E such that if E∈E then ∀F∈P(E)[F∈E]
- hereditary ring generated by
subadditivity
outer measure, μ∗ (p42) - extended real valued, non-negative, monotone and countably subadditive set function on an hereditary σ-ring with μ∗(∅)=0
- Theorem: If μ is a measure on a ring R and if:
  - $\forall A\in\mathbf{H}(\mathcal{R})[\mu^*(A)=\text{inf}\{\sum^\infty_{n=1}\mu(A_n)\ \vert\ (A_n)_{n=1}^\infty\subseteq\mathcal{R} \wedge A\subseteq \bigcup^\infty_{n=1}A_n\}]$ then $\mu^*$ is an extension of $\mu$ to an outer measure on $\mathbf{H}(\mathcal{R})$
- $\mu^*$ is the outer measure induced by the measure $\mu$
μ∗-measurable - given an outer measure μ∗ on a hereditary σ-ring H a set A∈H is $\mu^*$ -measurable if:
- ∀B∈H[μ∗(B)=μ∗(A∩B)+μ∗(B∩A′)]
  - PROBLEM: How can we do complementation in a ring?

@@ Line 11: / Line 11: @@
 *** {{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?'''