Sigma Algebra

Let $X$ be a Set. Then a $\sigma$-algebra $F$ is a nonempty collection of Subsets of $X$ such that the following hold:

1. The Empty Set is in $F$.

2. If $A$ is in $F$, then so is the complement of $A$.

3. If $A_n$ is a Sequence of elements of $F$, then the Union of the $A_n$s is in $F$.

If $S$ is any collection of subsets of $X$, then we can always find a $\sigma$-algebra containing $S$, namely the Power Set of $X$. By taking the Intersection of all $\sigma$-algebras containing $S$, we obtain the smallest such $\sigma$-algebra. We call the smallest $\sigma$-algebra containing $S$ the $\sigma$-algebra generated by $S$.

See also Borel Sigma Algebra, Borel Space, Measurable Set, Measurable Space, Measure Algebra, Standard Space