  We study the quantum ($C^*$) convexity structure of normalized positive
operator valued measures (POVMs) on measurable spaces. In particular, it is
seen that unlike extreme points under classical convexity, $C^*$-extreme points
of normalized POVMs on countable spaces (in particular for finite sets) are
always spectral measures (normalized projection valued measures). More
generally it is shown that atomic $C^*$-extreme points are spectral. A
Krein-Milman type theorem for POVMs has also been proved. As an application it
is shown that a map on any commutative unital $C^*$-algebra with countable
spectrum (in particular ${\mathbb C}^n$) is $C^*$-extreme in the set of unital
completely positive maps if and only if it is a unital $*$-homomorphism.
