Difference between revisions of "Pre-image sigma-algebra"
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(Created page with "Let {{M|f:X\rightarrow X'}} and let {{M|\mathcal{A}'}} be a algebra}} on {{M|X'}}, we can define a sigma algebra on {{M|X}}, called {{M|\mathcal{A} }...") |
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| + | {{DISPLAYTITLE:Pre-image {{sigma|algebra}}}}{{:Pre-image sigma-algebra/Infobox}} | ||
| + | {{Stub page|Add to sigma-algebra index, link to other pages, general expansion}} | ||
| + | {{Refactor notice}} | ||
| + | ==[[Pre-image sigma-algebra/Definition|Definition]]== | ||
| + | {{:Pre-image sigma-algebra/Definition}} | ||
| + | '''Claim: ''' {{M|(X,\mathcal{A})}} is indeed a {{sigma|algebra}} | ||
| + | ==Proof of claims== | ||
| + | {{Begin Inline Theorem}} | ||
| + | '''Claim 1: ''' {{M|(X,\mathcal{A})}} is indeed a [[sigma-algebra|{{sigma|algebra}}]] | ||
| + | {{Begin Inline Proof}} | ||
| + | {{:Pre-image sigma-algebra/Proof of claim: it is a sigma-algebra}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | ==References== | ||
| + | <references/> | ||
| + | {{Measure theory navbox|plain}} | ||
| + | {{Definition|Measure Theory}} | ||
| + | |||
| + | =OLD PAGE= | ||
Let {{M|f:X\rightarrow X'}} and let {{M|\mathcal{A}'}} be a [[Sigma-algebra|{{sigma|algebra}}]] on {{M|X'}}, we can define a sigma algebra on {{M|X}}, called {{M|\mathcal{A} }}, by: | Let {{M|f:X\rightarrow X'}} and let {{M|\mathcal{A}'}} be a [[Sigma-algebra|{{sigma|algebra}}]] on {{M|X'}}, we can define a sigma algebra on {{M|X}}, called {{M|\mathcal{A} }}, by: | ||
* {{M|1=\mathcal{A}:=f^{-1}(\mathcal{A}'):=\left\{f^{-1}(A')\vert\ A'\in\mathcal{A}'\right\} }} | * {{M|1=\mathcal{A}:=f^{-1}(\mathcal{A}'):=\left\{f^{-1}(A')\vert\ A'\in\mathcal{A}'\right\} }} | ||
{{Todo|Measures Integrals and Martingales - page 16}} | {{Todo|Measures Integrals and Martingales - page 16}} | ||
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Revision as of 13:57, 18 March 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
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Contents
[hide]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
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
