Countable generating algebra
= Countable generating algebra
{title2=$\sigma(\mathcal A)=\mathcal B$}
A countable <algebra of sets> generating the intended sigma-algebra. Standard Borel spaces admit such algebras by taking finite Boolean combinations from a countable topological base. Finite linear combinations of its indicators are dense in $L^2$ for every Borel <probability measure>, by the <Monotone class theorem>. Countability permits a common full-measure convergence set for all test indicators.