  This article studies an integral representation of functionals of linear
growth on metric measure spaces with a doubling measure and a Poincar\'e
inequality. Such a functional is defined through relaxation, and it defines a
Radon measure on the space. For the singular part of the functional, we get the
expected integral representation with respect to the variation measure. A new
feature is that in the representation for the absolutely continuous part, a
constant appears already in the weighted Euclidean case. As an application we
show that in a variational minimization problem related to the functional,
boundary values can be presented as a penalty term.
