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If $\Lambda $ is a measure space, $u:\Lambda ^{m}\rightarrow \Bbb{R}$ is agiven 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 $N$-meanwith kernel $u,$ a terminology borrowed from $U$-statistics. Physicalpotentials for systems of particles are also defined by generalized means.  This paper investigates whether various measure-theoretic concepts forgeneralized $N$-means are equivalent to the analogous concepts for theirkernels: a.e. convergence of sequences, measurability, essential boundednessand integrability with respect to absolutely continuous probability measures.The answer is often, but not always, positive. This information is crucial insome problems addressing the existence of generalized means satisfying givenconditions, such as the classical Inverse Problem of statistical physics (inthe canonical ensemble).

Some measure-theoretic properties of generalized means