= Right-continuous adapted processes are progressively measurable
{title2=$X|_{[0,T]\times\Omega}\text{ is }\mathcal B([0,T])\otimes\mathcal F_T\text{-measurable}$}
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.
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