Difference between revisions of "Measurable map"
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| + | ==See also== | ||
| + | * [[Random Variable]] | ||
| + | * [[Probability space]] | ||
| + | * [[Measurable space]] | ||
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| + | ==References== | ||
| + | <references/> | ||
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
Revision as of 22:41, 2 May 2015
Contents
[hide]Definition
Let (X,A) and (X′,A′) be measurable spaces
Then a map T:X→X′ is called A/A′-measurable if
T−1(A′)∈A, ∀A′∈A′
Notation
A given a measure space (a measurable space equipped with a measure) (X,A,μ) with a measurable map on the following mean the same thing:
- T:(X,A,μ)→(X′,A′,ˉμ) (if (X′,A′) is also equipped with a measure)
- T:(X,A,μ)→(X′,A′)
- T:(X,A)→(X′,A′)
We would write T:(X,A,μ)→(X′,A′) simply to remind ourselves of the measure we are using, it is not important to the concept of the measurable map.
Motivation
From the topic of random variables - which a special case of measurable maps (where the domain can be equipped with a probability measure, a measure where X has measure 1).
Consider: X:(Ω,A,P)→(V,U), we know that given a U∈U that T−1∈A which means we can measure it using P, which is something we'd want to do.
[Expand]
Example using sum of two die RV
