Suppose we have a probability space $(\Omega, \mathscr{F}, P)$ along with a filtration $\{\mathscr{F}_t\}_{t\in [0,1]}$.
Let $f:[0,1] \times \Omega \to \mathbb{R}^d$ be a progressive and square integrable process on $[0,1] \times \Omega$.
i.e. for every $t \in [0,1]$ the mapping $f(s,w)| _{ [0,1] \times \Omega }$ is $\mathscr{B}([0,t]) \otimes \mathscr{F}_t$ measurable and $\int \limits_\Omega \int \limits_0^1 f(s,\omega)^2 \mathop{ds} \mathop{dP} =E\bigg( \int \limits_0^1 f(s,\omega)^2 \mathop{ds} \bigg) < \infty $.
I would like to justify that the process $g:[0,1] \times \Omega \to \mathbb{R}^d$ given by $$g(t,\omega) := \int\limits_0^t f(s, \omega) \mathop{ds}$$ is also a progressive process with respect to the same filtration.
This is what I have so far:
Firstly I abuse notation by writing $f:= f|_{ [0,t] \times \Omega }$ (doesn't effect the definition of $g(t,\omega)$). Clearly this new $f$ is progressive and square integrable on $[0,t] \times \Omega$. All progressive processes are adapted, so just as in the construction of the Ito integral we may approximate $f$ with elementary functions $$\phi_n(s,\omega)= \sum \limits_{i=0}^{n-1} \xi_i(\omega) \chi_{[t_i,t_{i+1})}, \text{ such that } \lim \limits_{n\to \infty} E \bigg( \int \limits_0^1 (\phi_n(s,\omega)-f(s,\omega))^2 \mathop{ds} \bigg) =0.$$
We can interpret this as a limit in the $L^2$ sense on the product space. $L^p$ convergence implies convergence in measure, which in turn implies there is a subsequence $\phi_{n_k}$ that converges pointwise a.e. to $f$ on $[0,t] \times \Omega$.
Define $$\eta_k (\omega):= \int \limits_0^t \phi_{n_k} (s,\omega) = \sum_i \xi_i(\omega) \Delta t_i $$ which is clearly $\mathscr{F}_t$ measurable by the definition of elementary functions.
I claim that $ \eta_k \to \int\limits_0^t f(s, \omega) \mathop{ds}=g(t,\cdot)$ pointwise a.e. on $\Omega$.
If this holds then $g(t,\cdot)$ will be $\mathscr{F}_t$ measurable, and so the process $g(t, \omega)$ is adapted to the filtration. It is not difficult to show that $g(t,\omega)$ has continuous paths, so it is trivially RCLL. Adapted and RCLL implies Progressive, and then we would be done.
This issue that I'm having is with verifying the claim. I would like to use one of the convergence theorems to bring the limit inside the integral, but I don't see how to do this given I know very little about the $\phi_{n_k}$'s.
Any help would be greatly appreciated.