= Sigma-algebra generated by a collection
{title2=$\sigma(\mathcal I)=\bigcap_{\mathcal A\supseteq\mathcal I,\ \mathcal A\text{ a sigma-algebra}}\mathcal A$}
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>.
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