This talk has three related parts. In the first we discuss peak sets, classical and noncommutative. These will be important particularly in the second part, where we describe some recent progress on (Arveson's) noncommutative Hardy spaces--for general von Neumann algebras (joint with Louis Labuschagne). We use Haagerup's reduction theory to generalize Ueda's peak set theorem and its several striking consequences such as uniqueness of predual, an F and M Riesz theorem, a Gleason-Whitney theorem, etc. The third part flows out of, but is partly independent of, the second, and is joint work with Nik Weaver. We discuss some aspects of quantum measure theory, and quantum cardinals. Some of the proofs make use of Farah and Weaver's theory of quantum filters to investigate states on von Neumann algebras which are not normal but have other natural continuity properties. These are then applied to characterize the von Neumann algebras for which Ueda's peak set theorem holds.