An application of functional equations for generating ε-invariant measures

Add time:09/10/2019    Source:sciencedirect.com

Let (X,A,μ) be a probability space and let S:X→X be a measurable transformation. Motivated by K. Nikodem's paper from 1991 published in the Czechoslovak Mathematical Journal [7], we concentrate on a functional equation generating measures that are absolutely continuous with respect to μ and ε-invariant under S. As a consequence of the investigation, we obtain a result on the existence and uniqueness of solutions φ∈L1([0,1]) of the functional equationφ(x)=∑n=1N|fn′(x)|φ(fn(x))+g(x), where g∈L1([0,1]) and f1,…,fN:[0,1]→[0,1] are functions satisfying some extra conditions.

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