Site projects:Measure theory plan

Overview

The plan is to stick pretty close to what Halmos does but introduce definitions found elsewhere, for example Measures, Integrals and Martingales introduces a pre-measure on an algebra of sets, and a measure on a [ilmath]\sigma[/ilmath]-algebra of sets. Halmos defines only measure and talks of wanting to extend it to a [ilmath]\sigma[/ilmath]-algebra but he defines these on a [ilmath]\sigma[/ilmath]-ring instead.

Inline with the Notes:Measure theory plan page we will use the terms and symbols which may be found on the bottom of this page.

Questions / Problems / Reminders

• A page (perhaps measure theory and sigma algebras notice will need to (have part of it) be transcluded onto each relevant page. Not sure on much more than this right now. It will have to:
  • Urge the reader to keep algebras and sigma-algebras at the forefront of their minds when dealing with *measures
  • Urge the reader to keep algebras and sigma-algebras at the forefront of their minds when considering the definition of measurable and measure spaces

Progress

Covers only pages that existed before the project began

• n=31

Done

n=6

• Algebra (disambiguation) - DONE - Alec (talk) 00:05, 2 April 2016 (UTC)
• Algebra of sets - DONE - entire page overhauled (hadn't been changed for over a year) Alec (talk) 18:46, 1 April 2016 (UTC)
• Measure - DONE - old page entirely migrated to new page, proper format, proper terminology Alec (talk) 22:50, 30 March 2016 (UTC)
• Pre-image sigma-algebra - DONE - Alec (talk) 18:39, 1 April 2016 (UTC)
• Pre-image sigma-algebra/Definition - DONE - Alec (talk) 18:39, 1 April 2016 (UTC)
• Pre-image sigma-algebra/Proof of claim: it is a sigma-algebra - DONE - Alec (talk) 18:39, 1 April 2016 (UTC)

Work required

n=0

Not done

n=25

• Borel sigma-algebra
• Borel sigma-algebra generated by
• Extended real value
• Generator (sigma-algebra)
• Lebesgue measure
• Measurable function
• Measurable map
• Measurable space
• Measure space
• Outer-measure
• Pre-measurable space
• Pre-measure work saved on Pre-measure/New page Alec (talk) 15:57, 27 March 2016 (UTC)
• Pre-measure space
• Pre-measure/Properties in common with measure
• Probability function
• Probability measure
• Probability space
• Ring generated by
• Ring of sets
• Sigma-algebra
• Sigma-algebra generated by
• Sigma-algebra/Definition
• Sigma-field
• Sigma-ring
• Trace sigma-algebra

Symbols & Terms

Terms

Term	Meaning	Comment
Pre-measure (positive)	What it is now, but from a ring instead of algebra
Measure (positive)	What it is now, but from a [ilmath]\sigma[/ilmath]-ring instead of a [ilmath]\sigma[/ilmath]-algebra

Symbols

Symbol	Meaning
[ilmath]\mathcal{A} [/ilmath]	an algebra or [ilmath]\sigma[/ilmath]-algebra of sets.
[ilmath]\mathcal{B} [/ilmath]	Anything to do with Borel
[ilmath]\mathcal{C} [/ilmath]	an arbitrary collection of subsets of a set
[ilmath]\mathcal{G} [/ilmath]	Any generator of an object
[ilmath]\mathcal{R} [/ilmath]	a ring or [ilmath]\sigma[/ilmath]-ring of sets.
Generated objects
[ilmath]\mathcal{B}(\cdot)[/ilmath]	Borel-sigma-algebra of a topological space
[ilmath]\sigma(\cdot)[/ilmath]	[ilmath]\sigma[/ilmath]-algebra generated by
[ilmath]\sigma_R(\cdot)[/ilmath]	[ilmath]\sigma[/ilmath]-ring generated by