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.
@article{1217884489,
author = {EPHREMIDZE, Lasha and FUJII, Nobuhiko},
title = {On the uniqueness of the one-sided maximal functions of Borel measures},
journal = {J. Math. Soc. Japan},
volume = {60},
number = {1},
year = {2008},
pages = { 695-717},
language = {en},
url = {http://dml.mathdoc.fr/item/1217884489}
}
EPHREMIDZE, Lasha; FUJII, Nobuhiko. On the uniqueness of the one-sided maximal functions of Borel measures. J. Math. Soc. Japan, Tome 60 (2008) no. 1, pp. 695-717. http://gdmltest.u-ga.fr/item/1217884489/