 Sigma-algebra generated by a collection

The sigma-algebra generated by a collection is the smallest sigma-algebra containing all sets in that collection. The displayed intersection is over a nonempty family, since the power set is one candidate. An intersection of sigma-algebras contains the whole space and is closed under complements and countable unions, proving it is a sigma-algebra and establishing minimality.

For example, the half-lines $(-\infty ,x]$ generate the Borel sigma-algebra on the real line. They are Borel, and conversely $(-\infty ,b)={\bigcup }_n(-\infty ,b-1/n]$ and $(a,b)=(-\infty ,b)\setminus (-\infty ,a]$ recover every open interval. Preimages of these half-lines similarly generate the sigma-algebra generated by a random variable.