M\"{o}bius random law and infinite rank-one maps

We prove that Sarnak's conjecture holds for any infinite measure symbolic rank-one map. We further extended Bourgain-Sarnak's result, which says that the M\"{o}bius function is a good weight for the ergodic theorem, to maps acting on $\sigma$-finite measure spaces. We also discuss and extend Bourgain's theorem by establishing that there is a class of maps for which the M\"{o}bius disjointness property holds for any continuous bounded function. Our proof allows us to obtain an extension of Bourgain's theorem on M\"{o}bius disjointness for bounded rank one maps and a simple and self-contained proof of this fact.