Sigma algebra

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 $\Omega$ Let $P=2^{\Omega }$ be its power set, i.e. set of all subsets of $\Omega$. Let F ⊆ P such that all the following conditions are satisfied:

1. $\varnothing \in \Omega .$
2. If $A\in F$ then $A^{c}\in F$
3. If $G_{i}\in F$ for $i=1,2,3,\dots$ then $\bigcup _{i=1}^{\infty }G_{i}\in F$

Example

Given the set $\Omega$={Red,Yellow,Green}

The power set $2^{\Omega }$ 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={A0, A1, A4, A5, A7}, a subset of $2^{\Omega }$.

Notice that the following is satisfied:

1. The empty set is in F.
2. The original set $\Omega$ is in F.
3. For any set in F, you'll find it's complement in F as well.
4. 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 $\Omega$.

External links

• Tutorial