Suppose we have a measurable space $(\Omega,\mathcal{F})$ and an $\mathbb{R}$-valued continuous-time (but not necessarily continuous) stochastic process $X$. $X$ is jointly measurable if it is measurable with respect to the product $\sigma$-algebra $\mathcal{B}(\mathbb{R}^+)\otimes \mathcal{F}$.
An example of a process which is not jointly measurable is $X(t,\omega)=\mathbf{1}(t\in V)$, where $V$ is some non-Borel-measurable set. All left or right continuous processes are jointly measurable.
As a matter of curiosity, I tried constructing a martingale which is not jointly measurable, but didn't get very far. How might I do it?
Thanks.