  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).
