Notes:Measure theory plan

Purpose

This document is the plan for the measure theory notation and development on this site.

Introduce ring of sets
PRE-MEASURE ( $\bar{\mu}$ ) - Introduce a (positive) extended real valued countably additive set function, $\bar{\mu}$ on that ring. This will be a pre-measure and these are easy to create (use Lebesgue measure as example) which is why they're the first step.
OUTER-MEASURE ( $\mu^*$ ) - a construct named because it measures from the outside of a set and comes down (the inf), this lets us "measure" on a power-set like construction (a hereditary $\sigma$ -ring) which contains every subset of every set in the ring, as well as being closed under countable union and set subtraction.
PROBLEM: Outer measures are only subadditive not additive so they're not really measures. Make sure this weakness is demonstrated.
We need to consider only the sets that have the property of dividing up every other set in the hereditary sigma-ring additively.
We then show this new structure is a ring
We then show this new structure is a $\sigma$ -ring
MEASURE ( $\mu$ ) - The restriction of the outer-measure, $\mu^*$ , $\mu$ to this $\sigma$ -ring is a measure, a pre-measure but on a $\sigma$ -ring (instead of just ring)
Show $\mu$ is countably additive

We have now constructed a measure on a $\sigma$ -ring, $\mu$ from a pre-measure on a ring, $\bar{\mu}$

Show that $\sigma_R(\mathcal{R})$ (the sigma-ring generated by) is inside the $\sigma$ -ring constructed from the outer-measure.
Conclude that the sets in $\mathcal{R}$ are in this new ring (trivial/definition) and the job is done, we have constructed a measure on $\sigma_R(\mathcal{R})$