# Measurable space

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The message provided is:
Lets get this measure theory stuff sorted. At least the skeleton
• I can probably remove the old page... it doesn't say anything different.

## 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

1. REDIRECT Pre-measurable space/Definition

## Notes

1. More neatly written perhaps:
• [ilmath]A\subseteq\mathcal{P}(X)[/ilmath]

# 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]