Definition

measurability w.r.t. a $\sigma$-field

If $X \in \mathcal{B}/\mathcal{B}'$, $\mathcal{F} \subset \mathcal{B}$ is a $\sigma$-field, and $\sigma(X) \subset \mathcal{F}$ then we say $X$ is measurable with respect to $\mathcal{F}$ and we write $X \in \mathcal{F}$.