Sigma algebra: Difference between revisions
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A '''sigma algebra''' is an advanced mathematical concept. It refers to a [[mathematical structure|formal mathematical structure]] intended among other things to provide a rigid basis for [[axiomatic probability theory]]. | A '''sigma algebra''' is an advanced mathematical concept. It refers to a [[mathematical structure|formal mathematical structure]] intended among other things to provide a rigid basis for [[axiomatic probability theory]]. | ||
== | ==Formal definition== | ||
==Example== | |||
Given the set <math>\Omega</math>={Red,Yellow,Green} | Given the set <math>\Omega</math>={Red,Yellow,Green} | ||
The [[power set]] <math>2^\Omega</math> | The [[power set]] <math>2^\Omega</math> is {A0,A1,A2,A3,A4,A5,A6,A7}, with | ||
*A0={} (The empty set} | *A0={} (The empty set} | ||
*A1={Green} | *A1={Green} | ||
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#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. | #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. | ||
Thus F is a '''sigma algebra''' over <math>\Omega</math>. | |||
Revision as of 11:24, 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.
Formal definition
Example
Given the set ={Red,Yellow,Green}
The power set is {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.
Thus F is a sigma algebra over .