All examples of measurable but not progressive processes I have ever seen seemed to be based on the huge difference between $\mathcal{F}$ and $\mathcal{F}_\infty$. Here is what I mean.

Consider Example 19.5 from *Counterexamples in Probability* by Stoyanov: let $X_t ( \omega ) = 1_{t = \omega}$ be a process relative to $(\mathbb{R}_+, \mathcal{F}, \mathcal{F}_t)$, where $\mathcal{F} = \mathcal{B} ( \mathbb{R}_+ )$, whereas $\mathcal{F}_t$ does not depend on $t$ and is generated by points of $\mathbb{R}^d$ (that is, $A \in \mathcal{F}_t$ iff $A$ or $A^c$ is countable). Then it is easy to see that

- $X^{-1} ( \{ 1 \} ) = \{ ( t, \omega ) \ \colon \ t = \omega \in \mathbb{R}_+ \}$ belongs to $\mathcal{B}(\mathbb{R}_+) \otimes \mathcal{F}$, which means that $X$ is measurable,
- $X_t$ is $\mathcal{F}_t$-adapted,
- $X^{-1} ( \{ 1 \} ) \cap [0, t] \times \Omega$ does not belong to $\mathcal{B}( [0, t] ) \otimes \mathcal{F}_t$, which means that $X$ is not progressive.

Essentially the same example one can find in *Capacités et processus stochastiques* by Claude Dellacherie (page 47):
or a slight variation thereof on p.47 here (Example 1.38). In the latter $\mathcal{F}$ is the Lebesgue sigma-algebra on $[0, 1]$ and $\mathcal{F}_t = \mathcal{L}_0$ is generated by the null-sets.

Both examples seem somewhat artificial, because they only work because $\mathcal{F}$ is much bigger and most importantly has little to do with $\mathcal{F}_t$.

**My question is**: are there examples of measurable but not progressive processes with respect to $\mathcal{F} = \mathcal{F}_\infty$?

Probabilities and Potential. The exact same definition in Sharpe (1988), Rao (1995), Chung & Walsh (2005) and others. As a matter of fact, I've never seen such definitions as you say. Do you mean $\mathbb{P} \{ t \mapsto X_t \ \text{is measurable} \} = 1$? $\endgroup$2more comments