Difference between revisions of "Measurable space"
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| + | {{Refactor notice|grade=A*|msg=Lets get this measure theory stuff sorted. At least the skeleton | ||
| + | * I can probably remove the old page... it doesn't say anything different.}} | ||
| + | __TOC__ | ||
| + | ==Definition== | ||
| + | Given a [[set]], {{M|X}}, and a [[sigma-algebra|{{sigma|algebra}}]], {{M|\mathcal{A}\in\mathcal{P}(\mathcal{P}(X))}}<ref group="Note">More neatly written perhaps: | ||
| + | * {{M|A\subseteq\mathcal{P}(X)}}</ref> then a ''measurable space''{{rMIAMRLS}}{{rAGTARAF}} is the [[tuple]]: | ||
| + | * {{M|(X,\mathcal{A})}} | ||
| + | This is not to be confused with a ''[[measure space]]'' which is a [[tuple|{{M|3}}-tuple]]: {{M|(X,\mathcal{A},\mu)}} where {{M|\mu}} is a [[measure]] on the ''measurable space'' {{M|(X,\mathcal{A})}} | ||
| + | ===[[Premeasurable space]]=== | ||
| + | {{:Premeasurable space/Definition}} | ||
| + | ==See also== | ||
| + | * [[Pre-measurable space]] | ||
| + | * [[Measure space]] | ||
| + | ** [[Measure]] | ||
| + | ** [[Measurable map]] | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
| + | ==References== | ||
| + | <references/> | ||
| + | {{Definition|Measure Theory}} | ||
| + | =OLD PAGE= | ||
==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]] | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
Latest revision as of 13:05, 2 February 2017
Grade: A*
This page is currently being refactored (along with many others)
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The message provided is:
The message provided is:
Lets get this measure theory stuff sorted. At least the skeleton
- I can probably remove the old page... it doesn't say anything different.
Contents
Definition
Given a set, [ilmath]X[/ilmath], and a [ilmath]\sigma[/ilmath]-algebra, [ilmath]\mathcal{A}\in\mathcal{P}(\mathcal{P}(X))[/ilmath][Note 1] then a measurable space[1][2] is the tuple:
- [ilmath](X,\mathcal{A})[/ilmath]
This is not to be confused with a measure space which is a [ilmath]3[/ilmath]-tuple: [ilmath](X,\mathcal{A},\mu)[/ilmath] where [ilmath]\mu[/ilmath] is a measure on the measurable space [ilmath](X,\mathcal{A})[/ilmath]
Premeasurable space
- REDIRECT Pre-measurable space/Definition
See also
Notes
- ↑ More neatly written perhaps:
- [ilmath]A\subseteq\mathcal{P}(X)[/ilmath]
References
- ↑ Measures, Integrals and Martingales - René L. Schilling
- ↑ A Guide To Advanced Real Analysis - Gerald B. Folland
OLD PAGE
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]
