 Assuming noncontextuality for POVMs is, however, controversial, as POVMs are not fundamental, and some authors defend that noncontextuality should be assumed only for the underlying von Neumann measurements.Gleason's theorem, in its original version, does not hold if the Hilbert space is defined over the rational numbers, i.e., if the components of vectors in the Hilbert space are restricted to be rational numbers, or complex numbers with rational parts.However, when the set of allowed measurements is the set of all POVMs, the theorem holds.