Integrable solutions of inhomogeneous refinement type equations on intervals

Given a probability measure $P$ on a $\sigma$-algebra of subsets of a set $\mathrm{\Omega }$, an interval $I\subset \mathbb{R}$, $g\in L^1(I)$, and a function $\phi \text{\hspace{0.17em}⁣}:I\times \mathrm{\Omega }\to I$ fulfilling some conditions we obtain results on the existence of solutions $f\in L^1(I)$ of the inhomogeneous refinement type equation $$f(x)={\int }_{\mathrm{\Omega }}\mid {\phi }_x^{\mathrm{\prime }}(x,\omega )\mid f(\phi (x,\omega ))dP(\omega )+g(x)\mathrm{.}$$