An application of medial limits to iterative functional equations

Abstract: Assume that $(\Omega,\mathcal A,P)$ is a probability space, $f\colon[0,1] \times \Omega\to[0,1]$ is a function such that $f(0,\omega)=0$, $f(1,\omega)=1$ for every $\omega\in\Omega$, $g\colon[0,1]\to\mathbb R$ is a bounded function such that $g(0)=g(1)=0$, and $a,b\in\mathbb R$. Applying medial limits we describe bounded solutions $\varphi\colon[0,1] \to \mathbb R$ of the equation \begin{equation*} \varphi(x) = \int_\Omega \varphi(f(x,\omega)) dP(\omega)+g(x) \end{equation*} satisfying the boundary conditions $\varphi(0)=a$ and $\varphi(1)=b$.