Definition 3

*Let X be a set and let $\mathfrak{L}$ be a set of subsets of X. $\mathfrak{L}$ is said to be a $\sigma$-algebra on X if the following conditions are satisfied:

a) The complement of every set of $\mathfrak{L}$ belongs to $\mathfrak{L}$.

b) Every countable intersection of sets of $\mathfrak{L}$ belongs to $\mathfrak{L}$.

If $\mathfrak{L}$ is a $\sigma$-algebra on X, every countable union of sets of $\mathfrak{L}$ belongs to $\mathfrak{L}$ (for the complement of this union is an intersection of sets of $\mathfrak{L}$).

The set $\mathfrak{B}(X)$ of all subsets of X is clearly a $\sigma$-algebra. Every intersection of $\sigma$-algebras on X is a $\sigma$-algebra on X. For any subset of $\mathfrak{B}(X)$, there is therefore a smallest $\sigma$-algebra containing $\mathfrak{F}$; it is called the $\sigma$-algebra generated by $\mathfrak{F}$.