Notes:Halmos measure theory skeleton

Skeleton

• 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, $\mathcal{E}$ such that if $E\in\mathcal{E}$ then $\forall F\in\mathcal{P}(E)[F\in\mathcal{E}]$
  • hereditary ring generated by
• subadditivity
• outer measure, $\mu^*$ (p42) - extended real valued, non-negative, monotone and countably subadditive set function on an hereditary $\sigma$-ring with $\mu^*(\emptyset)=0$
  • Theorem: If $\mu$ is a measure on a ring $\mathcal{R}$ 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 $\mu$
• $\mu^*$-measurable - given an outer measure $\mu^*$ on a hereditary $\sigma$-ring $\mathcal{H}$ a set $A\in\mathcal{H}$ is $\mu^*$-measurable if:
  • $\forall B\in\mathcal{H}[\mu^*(B)=\mu^*(A\cap B)+\mu^*(B\cap A')]$
    • PROBLEM: How can we do complementation in a ring?