Difference between revisions of "Pre-image sigma-algebra"
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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 [ilmath]\sigma[/ilmath]-algebra | |
| [math]\{f^{-1}(A')\ \vert\ A'\in\mathcal{A}'\}[/math] is a [ilmath]\sigma[/ilmath]-algebra on [ilmath]X[/ilmath] given a [ilmath]\sigma[/ilmath]-algebra [ilmath](X',\mathcal{A}')[/ilmath] and a map [ilmath]f:X\rightarrow X'[/ilmath]. |
Definition
Let [ilmath]\mathcal{A}'[/ilmath] be a [ilmath]\sigma[/ilmath]-algebra on [ilmath]X'[/ilmath] and let [ilmath]f:X\rightarrow X'[/ilmath] be a map. The pre-image [ilmath]\sigma[/ilmath]-algebra on [ilmath]X[/ilmath][1] is the [ilmath]\sigma[/ilmath]-algebra, [ilmath]\mathcal{A} [/ilmath] (on [ilmath]X[/ilmath]) given by:
- [math]\mathcal{A}:=\left\{f^{-1}(A')\ \vert\ A'\in\mathcal{A}'\right\}[/math]
We can write this (for brevity) alternatively as:
- [math]\mathcal{A}:=f^{-1}(\mathcal{A}')[/math] (using abuses of the implies-subset relation)
Claim: [ilmath](X,\mathcal{A})[/ilmath] is indeed a [ilmath]\sigma[/ilmath]-algebra
Proof of claims
Claim 1: [ilmath](X,\mathcal{A})[/ilmath] is indeed a [ilmath]\sigma[/ilmath]-algebra
The message provided is:
References
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OLD PAGE
Let [ilmath]f:X\rightarrow X'[/ilmath] and let [ilmath]\mathcal{A}'[/ilmath] be a [ilmath]\sigma[/ilmath]-algebra on [ilmath]X'[/ilmath], we can define a sigma algebra on [ilmath]X[/ilmath], called [ilmath]\mathcal{A} [/ilmath], by:
- [ilmath]\mathcal{A}:=f^{-1}(\mathcal{A}'):=\left\{f^{-1}(A')\vert\ A'\in\mathcal{A}'\right\}[/ilmath]
TODO: Measures Integrals and Martingales - page 16
