Difference between revisions of "Pre-image sigma-algebra"
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{{DISPLAYTITLE:Pre-image {{sigma|algebra}}}}{{:Pre-image sigma-algebra/Infobox}} | {{DISPLAYTITLE:Pre-image {{sigma|algebra}}}}{{:Pre-image sigma-algebra/Infobox}} | ||
| − | {{Stub page|Add to sigma-algebra index, link to other pages, general expansion}} | + | {{Stub page|Add to sigma-algebra index, link to other pages, general expansion. Needs to be exemplary as a lot of search traffic enters here.|grade=A}} |
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==[[Pre-image sigma-algebra/Definition|Definition]]== | ==[[Pre-image sigma-algebra/Definition|Definition]]== | ||
{{:Pre-image sigma-algebra/Definition}} | {{:Pre-image sigma-algebra/Definition}} | ||
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==Proof of claims== | ==Proof of claims== | ||
{{Begin Inline Theorem}} | {{Begin Inline Theorem}} | ||
| − | '''Claim 1: ''' {{M|(X,\mathcal{A})}} is indeed a [[sigma-algebra|{{sigma|algebra}}]] | + | '''[[Pre-image sigma-algebra/Proof of claim: it is a sigma-algebra|Claim 1]]: ''' {{M|(X,\mathcal{A})}} is indeed a [[sigma-algebra|{{sigma|algebra}}]] |
{{Begin Inline Proof}} | {{Begin Inline Proof}} | ||
{{:Pre-image sigma-algebra/Proof of claim: it is a sigma-algebra}} | {{:Pre-image sigma-algebra/Proof of claim: it is a sigma-algebra}} | ||
{{End Proof}}{{End Theorem}} | {{End Proof}}{{End Theorem}} | ||
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==See also== | ==See also== | ||
* [[Trace sigma-algebra|Trace {{sigma|algebra}}]] | * [[Trace sigma-algebra|Trace {{sigma|algebra}}]] | ||
Latest revision as of 22:12, 19 April 2016
| Pre-image σ-algebra | |
| {f−1(A′) | A′∈A′} is a σ-algebra on X given a σ-algebra (X′,A′) and a map f:X→X′. |
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Add to sigma-algebra index, link to other pages, general expansion. Needs to be exemplary as a lot of search traffic enters here.
Grade: A
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Definition
Let A′ be a σ-algebra on X′ and let f:X→X′ be a map. The pre-image σ-algebra on X[1] is the σ-algebra, A (on X) given by:
- A:={f−1(A′) | A′∈A′}
We can write this (for brevity) alternatively as:
- A:=f−1(A′) (using abuses of the implies-subset relation)
Claim: (X,A) is indeed a σ-algebra
Proof of claims
See also
References
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OLD PAGE
Let f:X\rightarrow X' and let \mathcal{A}' be a \sigma-algebra on X', we can define a sigma algebra on X, called \mathcal{A} , by:
- \mathcal{A}:=f^{-1}(\mathcal{A}'):=\left\{f^{-1}(A')\vert\ A'\in\mathcal{A}'\right\}
TODO: Measures Integrals and Martingales - page 16
