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

Let $(X,\mathcal{A},\mu )$ be a probability space and let $S\text{\hspace{0.17em}⁣}:X\to X$ 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 $\varphi\in L^1([0,1])$ofthefunctionalequation$$\phi (x)={\sum }_{n=1}^N\mathrm{\mid }{f}_n^{\mathrm{\prime }}(x)\mathrm{\mid }\phi (f_n(x))+g(x),$$where$g\in L^1([0,1])$and$f_1,\ldots,f_N\colon[0,1]\to[0,1]$ are functions satisfying some extra conditions.