An application of functional equations for generating \(\varepsilon\)-invariant measures (Q2633878)

Let $g\in L^1[0,1]$ and $f_n:[0,1]\to [0,1]$ be given functions. The author solves the iterative functional equation $\varphi(x)=\sum_{n=1}^N |f'_n(x)|\varphi(f_n(x))+ g(x)$ in $L^1[0,1]$. It is assumed that strictly increasing contractions $f_n$ satisfy the equality $\sum_{n=1}^N |f'_n(x)|=1$ almost everywhere. The functional equation has a unique solution found by successive approximations.