I am interested to know if Ito integrals against Brownian motion can also be constructed via Skorohod representation. By this I mean the following: let $S_n$ be a simple random walk started at zero; for convenience assume the process linearly interpolates between times. Let $X^n$ be the process $$t \mapsto X^n_t = \frac{S_{nt}}{\sqrt{n}},$$ and of course the $X^n$ process converges weakly to Brownian motion (using the standard sup norm on $[0,1]$).

Now by Skorohod representation there exists a probabilty space supporting all the $X^n$ processes and a Brownian motion $B$ such that $X^n \to B$ almost surely. My question is if it is true that $$\int_0^1 f(t) dX^n_t \xrightarrow{a.s.} \int_0^1 f(t) dB_t$$ as $n \to \infty$, where $f$ is an $L^2$ function (deterministic even). The stochastic integral on the right is defined in the usual Ito type way as a limit in $L^2$ of the probability space.

It seems obvious that it is true for $f$ being an indicator function, and then by a density argument it goes through for general $f$. However I am a bit worried that I am missing something here and am looking for any insights. If it is true then that begs the question of whether or not it holds simultaneously for all $f$ at the same time, i.e. there is or is not an issue with null sets. Unfortunately null sets have a way of reducing me to a puddle of helplessness :)