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
