Sigma algebra: Difference between revisions
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imported>Ragnar Schroder (Initial stub) |
imported>Ragnar Schroder (Correcting tex, cosmetics, rephrasing,adding external link) |
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*A7={Red, Yellow, Green} (the whole set \Omega) | *A7={Red, Yellow, Green} (the whole set \Omega) | ||
Let F be a subset of 2^\Omega: F={A0, A1, A4, A5, A7}. | Let F be a subset of <math>2^\Omega</math>: F={A0, A1, A4, A5, A7}. | ||
Notice that the following is satisfied: | Notice that the following is satisfied: | ||
#The empty set is in F. | #The empty set is in F. | ||
#The original set <math>\Omega</math> is in F. | #The original set <math>\Omega</math> is in F. | ||
#For any set in F, you'll find it's [[complimentary set|complement]] | #For any set in F, you'll find it's [[complimentary set|complement]] in F as well. | ||
#For any subset of F, the union of the sets therein will also be in F. For example, the union of all elements in the subset {A0,A1,A4} of F is A0 | #For any subset of F, the union of the sets therein will also be in F. For example, the union of all elements in the subset {A0,A1,A4} of F is A0 U A1 U A4 = A5. | ||
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== External links == | == External links == | ||
*[http://www.probability.net/WEBdynkin.pdf| tutorial on www.probability.net] | |||
[[Category:Mathematics Workgroup]] | [[Category:Mathematics Workgroup]] | ||
Revision as of 12:21, 27 June 2007
A sigma algebra is an advanced mathematical concept. It refers to a formal mathematical structure intended among other things to provide a rigid basis for axiomatic probability theory.
Examples
Given the set ={Red,Yellow,Green}
The power set will be {A0,A1,A2,A3,A4,A5,A6,A7}, with
- A0={} (The empty set}
- A1={Green}
- A2={Yellow}
- A3={Yellow, Green}
- A4={Red}
- A5={Red, Green}
- A6={Red, Yellow}
- A7={Red, Yellow, Green} (the whole set \Omega)
Let F be a subset of : F={A0, A1, A4, A5, A7}.
Notice that the following is satisfied:
- The empty set is in F.
- The original set is in F.
- For any set in F, you'll find it's complement in F as well.
- For any subset of F, the union of the sets therein will also be in F. For example, the union of all elements in the subset {A0,A1,A4} of F is A0 U A1 U A4 = A5.