Does anyone know any sufficient conditions to put the limit into $d(\cdots)$? Namely, we want to realize: $$ \lim_{K\rightarrow\infty}\int_0^T f_t(\omega) \, d\left(\sum_{k=1}^K B_s^k(\omega)\right) = \int_0^T f_t(\omega) \, d\left(\lim_{K\rightarrow\infty} \sum_{k=1}^K B_s^k(\omega)\right) $$ where, we assume, $f_t$ enjoys integrability condition and $(B_s^k)_k$ is a sequence of one dimensional Brownian motion.
I know that stochastic integral enjoys bilinear (with respect to integrand and integrator). Hence, for a finite sum, we have:
$$
\sum_{k=1}^K \int_0^T f_t(\omega)\,d(B_s^k(\omega)) = \int_0^T f_t(\omega)\,d\left(\sum_{k=1}^K B_s^k(\omega)\right)
$$
But I totally have no idea about the infinite sum...and my professor also did not the sufficient condition to put the limit into $d(\cdots)$.