# 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

n=6

n=0

n=25

## 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