We introduce the notion of uniform γ-radonification of a family of operators, which unifies the notions of R-boundedness of a family of operators and γ-radonification of an individual operator. We study the properties of uniformly γ-radonifying families of operators in detail and apply our results to the stochastic abstract Cauchy problem Here, A is the generator of a strongly continuous semigroup of operators on a Banach space E, B is a bounded linear operator from a separable Hilbert space H

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... nto E, and W H is an H-cylindrical Brownian motion. When A and B are simultaneously diagonalisable, we prove that an invariant measure exists if and only if the family is uniformly γ-radonifying for some/all 0 < ϑ < π 2 , where S ϑ is the open sector of angle ϑ in the complex plane. This result can be viewed as a partial solution of a stochastic version of the Weiss conjecture in linear systems theory.