Notes:Measure theory plan

Purpose

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

Plan

• Introduce ring of sets
• PRE-MEASURE ([ilmath]\bar{\mu} [/ilmath]) - Introduce a (positive) extended real valued countably additive set function, [ilmath]\bar{\mu} [/ilmath] 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 ([ilmath]\mu^*[/ilmath]) - 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 [ilmath]\sigma[/ilmath]-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 [ilmath]\sigma[/ilmath]-ring
• MEASURE ([ilmath]\mu[/ilmath]) - The restriction of the outer-measure, [ilmath]\mu^*[/ilmath], [ilmath]\mu[/ilmath] to this [ilmath]\sigma[/ilmath]-ring is a measure, a pre-measure but on a [ilmath]\sigma[/ilmath]-ring (instead of just ring)
• Show [ilmath]\mu[/ilmath] is countably additive

We have now constructed a measure on a [ilmath]\sigma[/ilmath]-ring, [ilmath]\mu[/ilmath] from a pre-measure on a ring, [ilmath]\bar{\mu} [/ilmath]

Remaining steps

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