  Given a probability measure $P$ on a $\sigma$-algebra of subsets of a set
$\Omega$, an interval $I\subset\mathbb R$, $g\in L^1(I)$, and a function
$\varphi\colon I\times\Omega\to I$ fulfilling some conditions we obtain results
on the existence of solutions $f\in L^1(I)$ of the inhomogeneous refinement
type equation $$
f(x)=\int_{\Omega}\big|\varphi'_x(x,\omega)\big|f(\varphi(x,\omega))dP(\omega)+g(x).
$$
