An application of functional equations for generating $\epsilon$-invariant measures

Abstract

Let $(X,\mathcal{A},\mu )$ be a probability space and let $S:X\to X$ be a measurable transformation. Motivated by the paper of K. Nikodem [Czechoslovak Math. J. 41(116) (4) (1991) 565--569], we concentrate on a functional equation generating measures that are absolutely continuous with respect to $\mu$ and $\epsilon$-invariant under $S$. As a consequence of the investigation, we obtain a result on the existence and uniqueness of solutions $\phi \in L^1([0,1])$ of the functional equation $$\phi (x)=\sum _{n=1}^N|f_n^{\prime }(x)|\phi (f_n(x))+g(x),$$ where $g\in L^1([0,1])$ and $f_1,\dots ,f_N:[0,1]\to [0,1]$ are functions satisfying some extra conditions.