Measurability of stochastic processes on topological probability spaces

Abstract

A standard technique for converting a stochastic process ${\left(X_t\right)}_{t\in T}$ based on a probability space $(Y,\nu )$ into a separable process is to apply a lifting ρ for $L^{\infty }\left(\nu \right)$ to get the modified process ${\left(\rho \right(X_t\left)\right)}_{t\in T}$. In general the modified process is not measurable even if the original process is. We identify the liftings for which the modification always remains measurable as those which are 2-marginals.

Our focus is on the topological probability spaces Y which are Stone spaces of measure algebras. These have a canonical lifting σ which selects the clopen representative from each measure class. We analyze the marginal properties of σ by showing that for any probability space Ω, there is a thick set $Q\subseteq Y$ such that the restriction $\sigma \upharpoonright Q$ is a 2-marginal for the complete product space $\Omega \times Q$. It is known from a result of Musiał, Strauss, and Macheras that σ is not a 2-marginal for the usual product measure on $Y\times Y$ when Y is non-atomic (so we cannot take $Q=Y$). They asked whether this holds also for the Radon product measure. We show that it does under the Continuum Hypothesis, for some Stone spaces, building on work of Dudley and Cohn.