Theorem

necessary sufficient conditions for measurability

Let $\mathcal{B}, \mathcal{B}'$ be $\sigma$-algebras on $\Omega, \Omega '$ respectively and let $X: \Omega \to \Omega '$. If $\mathcal{B} ' = \sigma(\mathcal{C} ')$ then $X$ is measurable if and only if $X^{-1}(\mathcal{C} ') \sub \mathcal{B}$.