# Predictable Process Question (Da Prato & Zabcyzk 2014)

I've been looking at the 2014 edition of Da Prato & Zabcyzk and the sections on predictable processes. In particular, in their Proposition 3.7 (ii), they assert that if $$\Phi$$ is adapted and stochastically continuous on $$[0,T]$$, then it has a predictable modification.

What confuses me here, is it's not clear what $$\Phi$$ is. In part (i), $$\Phi$$ is an $$L(U,H)$$ valued operator ($$U$$ and $$H$$ both separable Hilbert spaces). If this is the case for (ii) also, what is the topology they wish to use on the space of operators? If it is the Borel $$\sigma$$-algebra induced by the operator norm, won't that introduce separability issues?

If it is the strong operator topology, then I am a bit unsure of a key step in their proof. They first construct a sequence of predictable $$\Phi_m$$ approximations, and define the set $$A$$ to be points where $$\{\Phi_m\}$$ converge, asserting $$A$$ is predictable. I do not see how they get the $$A$$ to be in the predictable $$\sigma$$-algebra.