Generalized stochastic processes revisited

Abstract

The paper addresses the question whether a random functional, a map from a set $E$ into the space of real-valued measurable functions on a probability space, has a measurable version with values in ${\mathbb R}^E$. Similarly, one may ask whether linear random functionals have versions in the algebraic dual. Most importantly, it can be asked which locally convex topological vector spaces $E$ have the ``regularity property'' that any linear random functional on $E$ has a version with values in the dual $E'$, an important issue in the theory of generalized stochastic processes. It has been shown by It\^{o} and Nawata that this is the case when $E$ is nuclear. However, the question of uniqueness has only been partially answered. We build up a framework where these and related questions can be clarified in terms of spaces and mappings. We study classes of spaces $E$ (beyond nuclear spaces) with the said regularity property, prove a seemingly new uniqueness result and exhibit various examples and counterexamples.