Difference between revisions of "Notes:Measure theory plan"

Revision as of 11:37, 24 March 2016

Purpose

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

Plan

• 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, $\bar{\mu}$

Remaining steps

• 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 $\sigma_R(\mathcal{R})$