Integrable solutions of inhomogeneous refinement type equations on intervals

Given a probability measure $P$ on a $σ$-algebra of subsets of a set $Ω$, an interval $I\subset\mathbb R$, $g\in L^1(I)$, and a function $φ\colon I\timesΩ\to I$ fulfilling some conditions we obtain results on the existence of solutions $f\in L^1(I)$ of the inhomogeneous refinement type equation $f(x)=\int_Ω\big|φ'_x(x,ω)\big|f(φ(x,ω))dP(ω)+g(x).$

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