Given $L^2(\Omega, \mathcal{F}, \mu; \mathbb R^n)$, a $\sigma$-algebra $\mathcal{G} \leq \mathcal{F}$, a function $f \in L^2(\Omega, \mathcal{F}, \mu; \mathbb R^n)$ which is $\mathcal{G}$-measurable, and a mapping $T: L^2(\Omega, \mathcal{F}, \mu; \mathbb R^n) \to L^2(\Omega, \mathcal{F}, \mu; \mathbb R^n)$.
Under which conditions on $T$ is $T(f)$ again $\mathcal{G}$-measurable? Is continuity enough? Linearity?