A Radon–Nikodým theorem for Fréchet measures

Abstract

We apply results in operator space theory to the setting of multidimensional measure theory. Using the extended Haagerup tensor product of Effros and Ruan, we derive a Radon–Nikodým theorem for bimeasures and then extend the result to general Fréchet measures (scalar-valued polymeasures). We also prove a measure-theoretic Grothendieck inequality, provide a characterization of the injective tensor product of two spaces of Lebesgue integrable functions, and discuss the possibility of a bounded convergence theorem for Fréchet measures.