  The paper deals with the theory of inner (outer) capacities on locally
compact spaces with respect to general function kernels, the main emphasis
being placed on the establishment of alternative characterizations of inner
(outer) capacities and inner (outer) capacitary measures for arbitrary sets.
The analysis is substantially based on the close interaction between the theory
of capacities and that of balayage. As a by-product, we provide a rigorous
justification of Fuglede's theories of inner and outer capacitary measures and
capacitability (Acta Math., 1960). The results obtained are largely new even
for the logarithmic, Newtonian, Green, $\alpha$-Riesz, and $\alpha$-Green
kernels.
