  Given a probability space $(\Omega,{\mathcal A},P)$, random variables
$L,M\colon\Omega\to\mathbb R$ and $g\in L^1(\mathbb R)$ we obtain two
characterizations of these $f\in L^1(\mathbb R)$ which are solutions of the
inhomogeneous refinement equation with a random affine map of the form
$f(x)=\int_\Omega |L(\omega)|f(L(\omega)x-M(\omega))P(d\omega)+g(x)$.
