Solution to the integrated Cauchy functional equation on the whole line

Abstract

A general solution of the integrated Cauchy functional equation ^∞∫-_∞f(x+y)dμ(y)=f(x)a.e. for x6(-∞,∞) is obtained under the only restriction that f is a locally integrable positive function and μ is a σ-finite positive Borel measure on R. This problem can be solved by an application of a general theorem due to Deny based on Choquet theory. However, we provide an alternative approach to the problem by using the more familiar Krein-Milman theorem.