Right-continuous adapted processes are progressively measurable ($X{|}_{[0,T]\times \Omega }\text{ is }\mathcal{B}([0,T])\otimes {\mathcal{F}}_T\text{-measurable}$)

For an adapted process with pathwise right continuity, fix $T$ and approximate its time argument by the next endpoint of a deterministic partition of $[0,T]$. Every approximation is jointly measurable for the terminal product sigma-algebra because all sampled values are ${\mathcal{F}}_T$-measurable. Right continuity makes them converge pointwise, proving progressive measurability. Sampling from the right need not preserve adaptedness at intermediate times, but that is not required in this proof.