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submitted for publicationInternational audienceWe introduce the notion of uniform $\gamma$--radonification of a family of operators, which unifies the notions of $R$--boundedness of a family of operators and $\gamma$--radonification of an individual operator. We study the the properties of uniformly $\gamma$--radonifying families of operators in detail and apply our results to the stochastic abstract Cauchy problem $$ dU(t) = AU(t) dt + B dW(t), \quad U(0)=0. $$ 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 $H$ into $E$, and $W_H$ is an $H$--cylindrical Brownian motion

Uniformly γ-radonifying families of operators and the stochastic Weiss conjecture