= Sigma-algebra generated by a random variable
{title2=$\sigma(X)=\{X^{-1}(B):B\in\mathcal B(\mathbb R)\}$}
For a real <random variable> $X$ on a <probability space>, $\sigma(X)$ is the smallest <sigma-algebra> making $X$ a <measurable function> into the <Borel sigma-algebra> on $\mathbb R$. The collection of all <inverse images> of <Borel sets> is already a <sigma-algebra>, since <inverse images> preserve <complements> and <countable unions>; this proves the displayed formula without completing any <measure>.
Back to article page
