Difference between revisions of "Pre-image sigma-algebra"
m (→Proof of claims: Linked to subpage in theorem box) |
m |
||
| Line 1: | Line 1: | ||
{{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}} |
| − | {{Refactor notice}} | + | {{Refactor notice|grade=A}} |
==[[Pre-image sigma-algebra/Definition|Definition]]== | ==[[Pre-image sigma-algebra/Definition|Definition]]== | ||
{{:Pre-image sigma-algebra/Definition}} | {{:Pre-image sigma-algebra/Definition}} | ||
Latest revision as of 22:12, 19 April 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:
See also
References
| ||||||||||||||||||||
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
