  This article is the first in a series dealing with the thermodynamic
properties of quantum Coulomb systems.
  In this first part, we consider a general real-valued function $E$ defined on
all bounded open sets of $\R^3$. Our aim is to give sufficient conditions such
that $E$ has a thermodynamic limit. This means that the limit
$E(\Omega_n)|\Omega_n|^{-1}$ exists for all `regular enough' sequence
$\Omega_n$ with growing volume, $|\Omega_n|\to\ii$, and is independent of the
considered sequence.
  The sufficient conditions presented in our work all have a clear physical
interpretation. In the next paper, we show that the free energies of many
different quantum Coulomb systems satisfy these assumptions, hence have a
thermodynamic limit.
