Let $X$ be a separable real Hilbert space, $(\Omega,\mathcal{F})$ be a measurable space, and $x:\Omega\to X$ be a measurable map. Then for each $\omega\in \Omega$, we can define $T(\omega)=x(\omega)\otimes x(\omega)$, which is a rank-one operator on $X$ satisfying $T(\omega)=\langle u, x(\omega)\rangle x(\omega)$ for all $u\in X$.
Clearly $T(\omega)$ is an element of different spaces of operators, such as the space of trace class operators, the space of Hilbert Schmidt operators, and the space of bounded operators. The topologies become coarser and coarser. May I know whether there is any result on the Borel measurability of the map $\omega\to T(\omega)$, with respect to different topologies?