We prove that if and are arbitrary (signed) Borel measures (on the unit circle) such that for each , where is the one-sided maximal operator (without modulus in the definition), then . The proof is constructive and it shows how can be recovered from in the unique way.
"On the uniqueness of the one-sided maximal functions of Borel measures." J. Math. Soc. Japan 60 (3) 695 - 717, July, 2008. https://doi.org/10.2969/jmsj/06030695