I'm reading Da Prato & Zabcyzk (2014), and in their Proposition 3.7, they assert that if $\Phi$ is an adapted and stochastically continuous process taking values, I believe, in $L(U,H)$ ($U$ and $H$ separable Hilbert spaces), then it has a predictable version. I am presuming they are using the strong operator topology that is introduced in their Section 1.2 here.
I follow the idea of the proof, where they first construct $\Phi_m$, a sequence of piecewise constant (in $t$) approximations which are clearly predictable, and then show that the sequence can be used to construct a predictable limit, $\Psi$, which is a modification of $\Phi$.
Where I am stuck is the following. They let $A$ be the set of $(t,\omega)$ for which $\Phi_m(t,\omega)$ converges, and then define $$ \Psi(t,\omega) = \begin{cases} \lim_{m\to \infty} \Phi_m(t,\omega) & (t,\omega)\in A,\\ 0 & (t,\omega)\notin A. \end{cases} $$ They remark that $A$, and consequently $\Psi$, are thus both predictable. Seeing that $A$ is predictable is where I am a bit stuck.
If its predictable, the rest of the proof is a straightforward application Borel-Cantelli to conclude that $\Psi$ is a modification of $\Phi$.