Difference between revisions of "Measure space"
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the tuple {{M|(X,\mathcal{A} )}} are a [[Pre-measurable space|pre-measurable space]] | the tuple {{M|(X,\mathcal{A} )}} are a [[Pre-measurable space|pre-measurable space]] | ||
==See also== | ==See also== | ||
| − | * [[ | + | * [[Pre-measurable space]] |
* [[Measurable space]] | * [[Measurable space]] | ||
| + | * [[Probability space]] | ||
* [[Pre-measure]] | * [[Pre-measure]] | ||
* [[Measure]] | * [[Measure]] | ||
Latest revision as of 15:24, 21 July 2015
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
