  In the theory of inner and outer balayage of positive Radon measures on a
locally compact space $X$ to arbitrary $A\subset X$ with respect to suitable,
quite general function kernels, developed in a series of the author's recent
papers, we find conditions ensuring the validity of the integral
representations. The results thereby obtained do hold and seem to be largely
new even for several interesting kernels in classical and modern potential
theory, which looks promising for possible applications. As an example of such
applications, we analyze how the total mass of a measure varies under its
balayage with respect to fractional Green kernels.
