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]

Remaining problems

If [ilmath]\cdot[/ilmath] is some arbitrary elements of the powerset (so [ilmath]\cdot\subseteq\mathcal{P}(X)[/ilmath]) what letter to use, for example, [ilmath]f:\mathcal{A}\rightarrow\text{whatever} [/ilmath] suggests an algebra in place. What letter to use for "just an arbitrary collection of subsets" eg for use on additive set function

Symbols and terminology

Symbols of:
Measure Theory
(Conventions established on this site) Order of introduction
Systems of sets Collections of subsets of [ilmath]X[/ilmath]
[ilmath]\mathcal{R} [/ilmath]	Ring of sets
[ilmath]\mathcal{A} [/ilmath]	Algebra of sets
(UNDECIDED)	Arbitrary collection of subsets
[ilmath]\mathcal{S} [/ilmath]	"Measurable" sets of the Outer-measure
Measures
[ilmath]\bar{\mu}:\mathcal{R}\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} [/ilmath]	Pre-measure
[ilmath]\mu^*:\mathcal{P}(X)\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} [/ilmath]	Outer-measure
[ilmath]\tilde{\mu}:\mathcal{S}\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} [/ilmath]	Measure induced by the outer-measure
[ilmath]\mu:\sigma_R(\mathcal{R})\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} [/ilmath]	measure induced on the sigma ring generated by

• [ilmath]\mathcal{R} [/ilmath] - Ring of sets - basically as it currently is
  • [ilmath]\mathcal{A} [/ilmath] - Mention Algebra of sets
• [ilmath]\bar{\mu}:\mathcal{R}\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} [/ilmath] (positive) Pre-measure - use the symbol [ilmath]\bar{\mu} [/ilmath] instead of [ilmath]\mu[/ilmath]
• [ilmath]\mu^*:\mathcal{P}(X)\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} [/ilmath] - outer-measure
• [ilmath]\mathcal{S} [/ilmath] for the "outer-measurable sets" (and discussion of definition), proof is ring, proof is [ilmath]\sigma[/ilmath]-ring
• [ilmath]\tilde{\mu}:\mathcal{S}\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} [/ilmath] - induced measure on [ilmath]\mathcal{S} [/ilmath] (if needed)
• [ilmath]\mu:\sigma_R(\mathcal{R}):\sigma_R(\mathcal{R})\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} [/ilmath] - induced measure on the generated sigma ring.