Abstract

Let (X,A,P), (Y,B,Q) be two arbitrary probability spaces and P:={(A,Py):y∈Y} be a regular conditional probability (rcp) on A with respect to Q. Denote by R the skew product of P and Q determined by P on the product σ-algebra A⊗B and by R^ its completion. I prove that if (X,A,P) is separable in the Fréchet–Nikodým pseudo-metric, then the stochastic process {ξy:y∈Y} has an equivalent measurable modification if and only if it is measurable with respect to a certain particular σ-algebra larger than A⊗B. The theorem is a strong generalization of two earlier results of the author and coauthors, where it was only proved that a suitable class of liftings transfer a measurable process into a measurable process. It is known that not every process possesses an equivalent measurable modification. My approach is essentially different from the earlier trials. It reverts to an earlier paper of Talagrand, who proved the existence of an equivalent separable modification of a measurable process (in case of R=P×Q), provided Y is endowed with a separable pseudo-metric.

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