  Given a bounded open set $\Omega\subset \mathbb{R}^n$, we study sequences of
quadratic functionals on the Sobolev space $H^1_0(\Omega)$, perturbed by
sequences of bounded linear functionals. We prove that their $\Gamma$-limits,
in the weak topology of $H^1_0(\Omega)$, can always be written as the sum of a
quadratic functional, a linear functional, and a non-positive constant. The
classical theory of $G$- and $H$-convergence completely characterises the
quadratic and linear parts of the $\Gamma$-limit and shows that their
coefficients do not depend on $\Omega$. The constant, which instead depends on
$\Omega$ and will be denoted by $-\nu(\Omega)$, plays an important role in the
study of the limit behaviour of the energies of the solutions. The main result
of this paper is that, passing to a subsequence, we can prove that $\nu$
coincides with a non-negative Radon measure on a sufficiently large collection
of bounded open sets $\Omega$. Moreover, we exhibit an example that shows that
the previous result cannot be obtained for every bounded open set. The specific
form of this example shows that the compactness theorem for the localisation
method in $\Gamma$-convergence cannot be easily improved.
