Difference between revisions of "Notes:Measure theory plan"
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(Created page with "==Purpose== This document is the ''plan'' for the measure theory notation and development on this site. ==Plan== * Introduce ring of sets * '''PRE-MEASURE''' ({{M|\bar{\m...") |
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We have now constructed a ''measure'' on a {{sigma|ring}}, {{M|\mu}} from a ''pre-measure'' on a ring, {{M|\bar{\mu} }} | We have now constructed a ''measure'' on a {{sigma|ring}}, {{M|\mu}} from a ''pre-measure'' on a ring, {{M|\bar{\mu} }} | ||
| + | |||
| + | ==Remaining steps== | ||
| + | * Show that {{M|\sigma_R(\mathcal{R})}} (the sigma-ring generated by) is inside the {{sigma|ring}} constructed from the outer-measure. | ||
| + | * Conclude that the sets in {{M|\mathcal{R} }} are in this new ring (trivial/definition) and the job is done, we have constructed a measure on {{M|\sigma_R(\mathcal{R})}} | ||
| + | |||
| + | ==Remaining problems== | ||
| + | If {{M|\cdot}} is some arbitrary elements of the powerset (so {{M|\cdot\subseteq\mathcal{P}(X)}}) what letter to use, for example, {{M|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== | ||
| + | {{Infobox | ||
| + | |title=Symbols of: | ||
| + | |above=<span style="font-size:2em;">Measure Theory</span> | ||
| + | |subheader=(Conventions established on this site)<br/>Order of introduction | ||
| + | |group1=test | ||
| + | |header1=[[System of sets|Systems of sets]]<br/>Collections of subsets of {{M|X}} | ||
| + | |label1={{M|\mathcal{R} }} | ||
| + | |data1=[[Ring of sets]] | ||
| + | |label2={{M|\mathcal{A} }} | ||
| + | |data2=[[Algebra of sets]] | ||
| + | |label3=(UNDECIDED) | ||
| + | |data3=Arbitrary collection of subsets | ||
| + | |label4={{M|\mathcal{S} }} | ||
| + | |data4="Measurable" sets of the [[Outer-measure]] | ||
| + | |header5=Measures | ||
| + | |label5={{M|\bar{\mu}:\mathcal{R}\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} }} | ||
| + | |data5=[[Pre-measure]] | ||
| + | |label6={{M|\mu^*:\mathcal{P}(X)\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} }} | ||
| + | |data6={{nowrap|[[Outer-measure]]}} | ||
| + | |label7={{M|\tilde{\mu}:\mathcal{S}\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} }} | ||
| + | |data7=[[Measure induced by the outer-measure]] | ||
| + | |label8={{M|\mu:\sigma_R(\mathcal{R})\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} }} | ||
| + | |data8=[[measure induced on the sigma ring generated by]] | ||
| + | }} | ||
| + | * {{M|\mathcal{R} }} - [[Ring of sets]] - basically as it currently is | ||
| + | ** {{M|\mathcal{A} }} - Mention [[Algebra of sets]] | ||
| + | * {{M|\bar{\mu}:\mathcal{R}\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} }} (positive) [[Pre-measure]] - use the symbol {{M|\bar{\mu} }} ''instead of'' {{M|\mu}} | ||
| + | * {{M|\mu^*:\mathcal{P}(X)\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} }} - [[outer-measure]] | ||
| + | * {{M|\mathcal{S} }} for the "outer-measurable sets" (and discussion of definition), proof is ring, proof is {{sigma|ring}} | ||
| + | * {{M|\tilde{\mu}:\mathcal{S}\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} }} - induced measure on {{M|\mathcal{S} }} (if needed) | ||
| + | * {{M|\mu:\sigma_R(\mathcal{R}):\sigma_R(\mathcal{R})\rightarrow\mathbb{R}_{\ge0}\cup\{+\infty\} }} - induced measure on the generated sigma ring. | ||
| + | |||
| + | {{Notes|Measure Theory}} | ||
Latest revision as of 19:28, 24 May 2016
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.
