We work with respect to a complete filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})$.
Let $X$ be an adapted stochastic process which is right-continuous in probability and such that, for every $t\in{\mathbb R}_+$ the set $$\left\{\int_0^t1_A\,dX\colon A\textrm{ is elementary}\right\} $$
is bounded in probability. Then, $X$ has a cadlag version (meaning that there exists a process $Y$ cadlag which has the same sample paths as $X$ almost surely).
An elementary predictable set is a subset of ${\mathbb R}_+\times\Omega$ which is a finite union of sets of the form $\{0\}\times F$ for $F\in\mathcal{F}_0$ and $(s,t]\times F$ for nonnegative reals $s<t$ and $F\in\mathcal{F}_s$
I am looking for an academic reference or a direct proof of that statement.