= Almost sure path regularity does not ensure progressive measurability
Let a probability space have two sample points, one of probability zero and one of probability one, and give it the complete power-set sigma-algebra at every time. Let $A\subseteq[0,1]$ be non-Borel. Assign the <stochastic process> the path $\mathbf1_A(t)$ at the null sample point and the zero path at the other point. Each fixed-time variable is measurable and the paths are <càdlàg> <almost surely>, but the <stochastic process> is not <progressively measurable>: a section in the time variable of a product-measurable function must be Borel measurable at every sample point. The zero <stochastic process> is an <indistinguishable> <progressively measurable> version. Completing the filtration does not complete the time-product sigma-algebra or remove this distinction.
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