Difference between revisions of "Pre-image sigma-algebra"
From Maths
(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} }...") |
m |
||
| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| + | {{DISPLAYTITLE:Pre-image {{sigma|algebra}}}}{{:Pre-image sigma-algebra/Infobox}} | ||
| + | {{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|grade=A}} | ||
| + | ==[[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}} | ||
| + | '''[[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}} | ||
| + | {{:Pre-image sigma-algebra/Proof of claim: it is a sigma-algebra}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | |||
| + | ==See also== | ||
| + | * [[Trace sigma-algebra|Trace {{sigma|algebra}}]] | ||
| + | ==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}} | ||
| − | |||
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′. |
Stub grade: A
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
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
This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.
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
| ||||||||||||||||||||
OLD PAGE
Let f:X→X′ and let A′ be a σ-algebra on X′, we can define a sigma algebra on X, called A, by:
- A:=f−1(A′):={f−1(A′)| A′∈A′}
TODO: Measures Integrals and Martingales - page 16
