  The concept of boundary plays an important role in several branches of
general relativity, e.g., the variational principle for the Einstein equations,
the event horizon and the apparent horizon of black holes, the formation of
trapped surfaces. On the other hand, in a branch of mathematics known as
geometric measure theory, the usefulness has been discovered long ago of yet
another concept, i.e., the reduced boundary of a finite-perimeter set. This
paper proposes therefore a definition of finite-perimeter sets and their
reduced boundary in general relativity. Moreover, a basic integral formula of
geometric measure theory is evaluated explicitly in the relevant case of
Euclidean Schwarzschild geometry, for the first time in the literature. This
research prepares the ground for a measure-theoretic approach to several
concepts in gravitational physics, supplemented by geometric insight. Moreover,
such an investigation suggests considering the possibility that the in-out
amplitude for Euclidean quantum gravity should be evaluated over
finite-perimeter Riemannian geometries that match the assigned data on their
reduced boundary. As a possible application, an analysis is performed of the
basic formulae leading eventually to the corrections of the intrinsic quantum
mechanical entropy of a black hole.
