  Under suitable requirements on a kernel on a locally compact space, we
develop a theory of inner (outer) balayage of quite general Radon measures
$\omega$ (not necessarily of finite energy) onto quite general sets (not
necessarily closed). We prove the existence and the uniqueness of inner (outer)
swept measures, analyze their properties, and provide a number of alternative
characterizations. In spite of being in agreement with Cartan's theory of
Newtonian balayage, the results obtained require essentially new methods and
approaches, since in the case in question, useful specific features of
Newtonian potentials may fail to hold. The theory thereby established extends
considerably that by Fuglede (Anal. Math., 2016) and that by the author (Anal.
Math., 2022), these two dealing with $\omega$ of finite energy. Such a
generalization enables us to improve substantially our recent results on the
inner Gauss variational problem (Constr. Approx., 2023), by strengthening their
formulations and/or by extending the area of their validity. This study covers
many interesting kernels in classical and modern potential theory, which also
looks promising for other applications.
