Given measurable spaces $(\Omega_1, \mathcal{F}_1)$ and $(\Omega_2, \mathcal{F}_2)$, and the mapping $T: \Omega_1 \rightarrow \Omega_2$, $T$ is said to be $\mathcal{F}_1/\mathcal{F}_2$ measurable if $T^{-1}(B)\in \mathcal{F}_1$ for all $B \in \mathcal{F}_2$.
Can I claim that $T(A) \in \mathcal{F}_2$ for each $A \in \mathcal{F}_1$? If not, what is significant about the transformations which have/don't have this property?