  Defining a condenser in a locally compact space as a locally finite,
countable collection of Borel sets $A_i$, $i\in I$, with the sign $s_i=\pm1$
prescribed such that $A_i\cap A_j=\varnothing$ whenever $s_is_j=-1$, we
consider a minimum energy problem with an external field over infinite
dimensional vector measures $(\mu^i)_{i\in I}$, where $\mu^i$ is a suitably
normalized positive Radon measure carried by $A_i$ and such that
$\mu^i\leqslant\xi^i$ for all $i\in I_0$, $I_0\subset I$ and constraints
$\xi^i$, $i\in I_0$, being given. If $I_0=\varnothing$, the problem reduces to
the (unconstrained) Gauss variational problem, which is in general unsolvable
even for a condenser of two closed, oppositely signed plates in $\mathbb R^3$
and the Coulomb kernel. Nevertheless, we provide sufficient conditions for the
existence of solutions to the stated problem in its full generality, establish
the vague compactness of the solutions, analyze their uniqueness, describe
their weighted potentials, and single out their characteristic properties. The
strong and the vague convergence of minimizing nets to the minimizers is also
studied. The phenomena of non-existence and non-uniqueness of solutions to the
problem are illustrated by examples. The results obtained are new even for the
classical kernels on $\mathbb R^n$, $n\geqslant2$, and closed $A_i$, $i\in I$,
which is important for applications.
