Some measure-theoretic properties of generalized means

TL;DR

This paper explores measure-theoretic properties of generalized means, examining when properties like measurability and integrability of these means are equivalent to those of their kernels, with implications for statistical physics.

Contribution

It provides conditions under which measure-theoretic properties of generalized means and their kernels are equivalent, aiding in solving inverse problems in statistical physics.

Findings

Equivalence of a.e. convergence for means and kernels

Measurability of means and kernels are often equivalent

Integrability conditions are sometimes equivalent

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).