On the uniqueness of the one-sided maximal functions of Borel measures

Abstract

We prove that if ν and μ are arbitrary (signed) Borel measures (on the unit circle) such that M₊ν(x) = M₊μ(x) for each x, where M₊ is the one-sided maximal operator (without modulus in the definition), then ν = μ. The proof is constructive and it shows how ν can be recovered from M₊ν in the unique way.