Some measure-theoretic properties of generalized means

Abstract

If $\Lambda $ is a measure space, $u:\Lambda ^{m}\rightarrow \Bbb{R}$ is a given function and $N\geq m,$ the function $U(x_{1},...,x_{N})=\left( \begin{array}{l} N \\ m \end{array} \right) ^{-1}\sum_{1\leq i_{1}<\cdots <i_{m}\leq N}u(x_{i_{1}},...,x_{i_{m}}) $ is called the generalized $N$-mean with kernel $u,$ a terminology borrowed from $U$-statistics. Physical potentials for systems of particles are also defined by generalized means. This paper investigates whether various measure-theoretic concepts for generalized $N$-means are equivalent to the analogous concepts for their kernels: a.e. convergence of sequences, measurability, essential boundedness and integrability with respect to absolutely continuous probability measures. The answer is often, but not always, positive. This information is crucial in some problems addressing the existence of generalized means satisfying given conditions, such as the classical Inverse Problem of statistical physics (in the canonical ensemble).