Difference between revisions of "Measurable space"
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==Definition== | ==Definition== | ||
| − | + | A ''measurable space''<ref name="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> is a [[Tuple|tuple]] consisting of a set {{M|X}} and a [[Sigma-algebra|{{Sigma|algebra}}]] {{M|\mathcal{A} }}, which we denote: | |
| − | + | * {{M|(X,\mathcal{A})}} | |
| + | ==Pre-measurable space== | ||
| + | A ''pre-measurable space''<ref name="ALEC">Alec's own terminology, it's probably not in books because it's barely worth a footnote</ref> is a set {{M|X}} coupled with an [[Algebra of sets|algebra]], {{M|\mathcal{A} }} (where {{M|\mathcal{A} }} is '''NOT''' a {{sigma|algebra}}) which we denote as follows: | ||
| + | * {{M|(X,\mathcal{A})}} | ||
==See also== | ==See also== | ||
| + | * [[Pre-measure space]] | ||
* [[Measure space]] | * [[Measure space]] | ||
* [[Measurable map]] | * [[Measurable map]] | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
Revision as of 15:27, 21 July 2015
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]
