Some measure-theoretic properties of U-statistics applied in statistical physics

Abstract

This paper investigates the relationship between various measure-theoretic properties of U-statistics with fixed sample size $N$ and the same properties of their kernels. Specifically, the random variables are replaced with elements in some measure space $(\Lambda; dx)$, the resultant real-valued functions on $\Lambda^N$ being called generalized $N$-means. It is shown that a.e. convergence of sequences, measurability, essential boundedness and, under certain conditions, integrability with respect to probability measures of generalized $N$-means and their kernels are equivalent. These results are crucial for the solution of the inverse problem in classical statistical mechanics in the canonical formulation.