# Measurable space

The message provided is:

- I can probably remove the old page... it doesn't say anything different.

## Contents

## Definition

Given a set, [ilmath]X[/ilmath], and a [ilmath]\sigma[/ilmath]-algebra, [ilmath]\mathcal{A}\in\mathcal{P}(\mathcal{P}(X))[/ilmath]^{[Note 1]} then a *measurable space*^{[1]}^{[2]} is the tuple:

- [ilmath](X,\mathcal{A})[/ilmath]

This is not to be confused with a *measure space* which is a [ilmath]3[/ilmath]-tuple: [ilmath](X,\mathcal{A},\mu)[/ilmath] where [ilmath]\mu[/ilmath] is a measure on the *measurable space* [ilmath](X,\mathcal{A})[/ilmath]

### Premeasurable space

- REDIRECT Pre-measurable space/Definition

## See also

## Notes

- ↑ More neatly written perhaps:
- [ilmath]A\subseteq\mathcal{P}(X)[/ilmath]

## References

- ↑ Measures, Integrals and Martingales - René L. Schilling
- ↑ A Guide To Advanced Real Analysis - Gerald B. Folland

# OLD PAGE

## Definition

A *measurable space*^{[1]} is a tuple consisting of a set [ilmath]X[/ilmath] and a [ilmath]\sigma[/ilmath]-algebra [ilmath]\mathcal{A} [/ilmath], which we denote:

- [ilmath](X,\mathcal{A})[/ilmath]

## Pre-measurable space

A *pre-measurable space*^{[2]} is a set [ilmath]X[/ilmath] coupled with an algebra, [ilmath]\mathcal{A} [/ilmath] (where [ilmath]\mathcal{A} [/ilmath] is **NOT** a [ilmath]\sigma[/ilmath]-algebra) which we denote as follows:

- [ilmath](X,\mathcal{A})[/ilmath]