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 ($\bar{\mu}$) - 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 $\sigma_R(\mathcal{R})$

Remaining problems

If $\cdot$ is some arbitrary elements of the powerset (so $\cdot\subseteq\mathcal{P}(X)$) what letter to use, for example, $f:\mathcal{A}\rightarrow\text{whatever}$ 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 $X$
$\mathcal{R}$	Ring of sets
$\mathcal{A}$	Algebra of sets
(UNDECIDED)	Arbitrary collection of subsets
$\mathcal{S}$	"Measurable" sets of the Outer-measure
Measures
$\bar{\mu}:\mathcal{R}\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\}$	Pre-measure
$\mu^*:\mathcal{P}(X)\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\}$	Outer-measure
$\tilde{\mu}:\mathcal{S}\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\}$	Measure induced by the outer-measure
$\mu:\sigma_R(\mathcal{R})\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\}$	measure induced on the sigma ring generated by

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