# Measure space

From Maths

**Note:** This page requires knowledge of measurable spaces.

## Definition

A *measure space*^{[1]} is a tuple:

- [ilmath](X,\mathcal{A},\mu:\mathcal{A}\rightarrow[0,+\infty])[/ilmath] - but because Mathematicians are lazy we simply write:
- [math](X,\mathcal{A},\mu)[/math]

Where [ilmath]X[/ilmath] is a set, and [ilmath]\mathcal{A} [/ilmath] is a [ilmath]\sigma[/ilmath]-algebra on that set (which together, as [ilmath](X,\mathcal{A})[/ilmath], form a measurable space) and [ilmath]\mu [/ilmath] is a measure.

## Pre-measure space

Given a set [ilmath]X[/ilmath] and an algebra, [ilmath]\mathcal{A} [/ilmath] (NOT a [ilmath]\sigma[/ilmath]-algebra) we can define a *pre-measure space*^{[2]} as follows:

- [ilmath](X,\mathcal{A},\mu_0)[/ilmath] where [ilmath]\mu_0[/ilmath] is a Pre-measure (a mapping, [ilmath]\mu_0:\mathcal{A}\rightarrow[0,+\infty][/ilmath] with certain properties)

the tuple [ilmath](X,\mathcal{A} )[/ilmath] are a pre-measurable space