Sigma algebra: Difference between revisions

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imported>Michael Hardy
imported>Aleksander Stos
(→‎Example: this was false (replaced by a trivial example))
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# If <math>G_i \in F</math> for <math>i = 1,2,3,\dots</math> then  <math>\bigcup_{i =1}^{\infty} G_{i} \in F </math>
# If <math>G_i \in F</math> for <math>i = 1,2,3,\dots</math> then  <math>\bigcup_{i =1}^{\infty} G_{i} \in F </math>


==Example==
== Example ==
Given the set <math>\Omega</math>={Red,Yellow,Green}
The power set itself is a &sigma; algebra.
 
The [[power set]]  <math>2^\Omega</math> 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 <math>\Omega</math>)
 
Let F={A0, A1, A4, A5, A7}, a subset of <math>2^\Omega</math>.
 
Notice that the following is satisfied:
#The empty set 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]] 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 <math>\Omega</math>.


== See also ==
== See also ==

Revision as of 15:27, 10 July 2007

In mathematics, a sigma algebra is a formal mathematical structure intended among other things to provide a rigid basis for axiomatic probability theory.

Formal definition

Given a set Let be its power set, i.e. set of all subsets of . Let F ⊆ P such that all the following conditions are satisfied:

  2. If then
  3. If for then

Example

The power set itself is a σ algebra.

See also

References

External links

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