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

Abstract

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