  The paper deals with minimum energy problems in the presence of external
fields with respect to the Riesz kernels $|x-y|^{\alpha-n}$, $0<\alpha<n$, on
$\mathbb R^n$, $n\geqslant2$. For quite a general (not necessarily lower
semicontinuous) external field $f$, we obtain necessary and/or sufficient
conditions for the existence of $\lambda_{A,f}$ minimizing the Gauss functional
\[\int|x-y|^{\alpha-n}\,d(\mu\otimes\mu)(x,y)+2\int f\,d\mu\] over all positive
Radon measures $\mu$ with $\mu(\mathbb R^n)=1$, concentrated on quite a general
(not necessarily closed) $A\subset\mathbb R^n$. We also provide various
alternative characterizations of the minimizer $\lambda_{A,f}$, analyze the
continuity of both $\lambda_{A,f}$ and the modified Robin constant for monotone
families of sets, and give a description of the support of $\lambda_{A,f}$. The
significant improvement of the theory in question thereby achieved is due to a
new approach based on the close interaction between the strong and the vague
topologies, as well as on the theory of inner balayage, developed recently by
the author.
