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

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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.

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