Difference between revisions of "Measure space"
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| − | + | '''Note:''' This page requires knowledge of [[Measurable space|measurable spaces]]. | |
==Definition== | ==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]]. | |
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| + | ==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== | ==See also== | ||
* [[Probability space]] | * [[Probability space]] | ||
Revision as of 15:18, 21 July 2015
Note: This page requires knowledge of measurable spaces.
Contents
[hide]Definition
A measure space[1] is a tuple:
- (X,A,μ:A→[0,+∞]) - but because Mathematicians are lazy we simply write:
- (X,A,μ)
Where X is a set, and A is a σ-algebra on that set (which together, as (X,A), form a measurable space) and μ is a measure.
Pre-measure space
Given a set X and an algebra, A (NOT a σ-algebra) we can define a pre-measure space[2] as follows:
- (X,A,μ0) where μ0 is a Pre-measure (a mapping, μ0:A→[0,+∞] with certain properties)
the tuple (X,A) are a pre-measurable space
