  A second, independent line of thought leading to the KS theorem is provided by Gleason’s theorem (Gleason 1957). The theorem states that on a Hilbert space of dimension greater than or equal to 3, the only possible probability measures are the measures μ(Pα) = Tr(Pα W), where Pα is a projection operator, W is the statistical operator characterizing the system’s actual state and Tr is the trace   operation.[4]  The Pα can be understood as representing yes-no observables, i.e. questions for whether a QM system ‘living’ in such a Hilbert space has a property α or not, and every possible property α is associated uniquely with a vector |α&gt; in the space — so, the task is to unambiguously assign probabilities to all vectors in the space. Now, the QM measure μ is continuous, so Gleason’s theorem in effect proves that every probability assignment to all the possible properties in a three-dimensional Hilbert space must be continuous, i.e. must map all vectors in the space continuously into the interval [0, 1]. On the other hand, an HV theory (if characterized by VD + NC) would imply that of every property we can say whether the system has it or not. This yields a trivial probability function which maps all the Pα to either 1 or 0, and, provided that values 1 and 0 both occur (which follows trivially from interpreting the numbers as probabilities), this function must clearly be discontinuous (cf. Redhead 1987: 28).