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

$C^*$-extreme points of positive operator valued measures and unital completely positive maps