sigma-algebra

Consider a set \[X\]. A sigma-algebra \[\mathcal{F}\] of subsets of \[X\] is a collection \[\mathcal{F}\] of subsets of \[X\] satisfying the following conditions:

1. \[\emptyset\in \mathcal{F}\]
   
2. If \[B\in \mathcal{F}\] then its complement (elements not in \[B\]) \[B^{c}\] is also in \[\mathcal{F}\]
   
3. If \[B_{1},B_{2},\dots\] is a countable collection of sets in \[\mathcal{F}\], then the set which is their union \[\cup_{n=1}^{\infty}B_{n}\] also belongs in \[\mathcal{F}\]
   

A simple example would be if \[X=\left\{ a,b,c,d \right\}\], one possible sigma-algebra on \[X\] is \[\mathcal{F}=\left\{ \emptyset,\left\{ a,b \right\},\left\{ c,d \right\},\left\{ a,b,c,d \right\} \right\}\]. One thing you would notice is that the sigma-algebras of subsets of some \[X\] always lies between two extremes, i.e. \[\left\{ \emptyset,X \right\}\subset \mathcal{F}\subset \mathcal{P}(X)\], where \[\left\{ \emptyset, X \right\}\] is the collection consisting of the empty set and \[X\] itself and \[\mathcal{P}(X)\] is every possible combination of subsets of \[X\]. Note that \[\mathcal{P}(X)\] is also called the power set on \[X\].