Difference between revisions of "Measure space"
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| − | + | '''Note:''' This page requires knowledge of [[Measurable space|measurable spaces]]. | |
| − | == | + | ==Definition== |
| − | A [[ | + | A ''measure space''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> is a [[Tuple|tuple]]: |
| + | * {{M|(X,\mathcal{A},\mu:\mathcal{A}\rightarrow[0,+\infty])}} - but because [[Mathematicians are lazy]] we simply write: | ||
| + | ** {{MM|(X,\mathcal{A},\mu)}} | ||
| + | Where {{M|X}} is a set, and {{M|\mathcal{A} }} is a [[Sigma-algebra|{{Sigma|algebra}}]] on that set (which together, as {{M|(X,\mathcal{A})}}, form a [[Measurable space|measurable space]]) and {{M|\mu }} is a [[Measure|measure]]. | ||
| − | < | + | ==Pre-measure space== |
| + | Given a set {{M|X}} and an [[Algebra of sets|algebra]], {{M|\mathcal{A} }} (NOT a {{sigma|algebra}}) we can define a ''pre-measure space''<ref name="ALEC">Alec's own terminology. It is likely not in books because it's barely worth a footnote</ref> as follows: | ||
| + | * {{M|(X,\mathcal{A},\mu_0)}} where {{M|\mu_0}} is a [[Pre-measure]] (a mapping, {{M|\mu_0:\mathcal{A}\rightarrow[0,+\infty]}} with certain properties) | ||
| + | the tuple {{M|(X,\mathcal{A} )}} are a [[Pre-measurable space|pre-measurable space]] | ||
| + | ==See also== | ||
| + | * [[Pre-measurable space]] | ||
| + | * [[Measurable space]] | ||
| + | * [[Probability space]] | ||
| + | * [[Pre-measure]] | ||
| + | * [[Measure]] | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
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
