  Under suitable requirements on a kernel on a locally compact space, we
develop a theory of inner (outer) pseudo-balayage of quite general signed Radon
measures (not necessarily of finite energy) onto quite general sets (not
necessarily closed). Such investigations were initiated in Fuglede's work
(Anal. Math., 2016), which was, however, mainly concerned with the outer
pseudo-balayage of positive measures of finite energy, whereas treatment of
signed measures of infinite energy requires essentially new methods and
approaches. To illustrate this, it is enough to note that in the particular
case of kernels satisfying the balayage principle, the pseudo-balayage of
positive measures coincides with their balayage, whereas for signed measures,
these two concepts are drastically different. The theory thereby established
enables us to provide necessary and sufficient conditions for the solvability
of the inner Gauss variational problem, a point of interest for a number of
recent researches in constructive function theory. This study covers many
interesting kernels in classical and modern potential theory, which looks
promising for further applications.
