For a real valued stochastic process there is predictable sigma algebra as is generated by all adapted left continuous processes, and the optional sigma algebra generated by adapted RCLL processes. I wonder if there is the notion of a sigma algebra generated by all adapted processes, by all progressive processes. If so, what are their characteristics, i.e., what are their typical class of generator sets? By generator sets I mean a simple class of sets generating these sigma-algebras.
Rogers and Williams (Diffusions, Markov processes and Martingales Vol. 2, p. 314) define the sigma-algebra generated by all progressive processes. In fact, they define a progressive set as a subset $F$ of $[0,\infty)\times\Omega$ of which the indicator function $1_F$ is a progressive process. The collection of progressive sets forms a sigma-algebra. They then say that ``a process is progressive if and only if it is measurable relative to this 'progressive $\sigma$-algebra'. You can easily prove this by monotone-class arguments.'' (Wanted to go over checking this myself, then I ran into your question.)