 Let formula_5 be a function from projection operators to the unit interval with the property that, if a set formula_6 of projection operators sum to the identity matrix (that is, if they correspond to an orthonormal basis), then

Such a function expresses an assignment of probability values to the outcomes of measurements, an assignment that is "noncontextual" in the sense that the probability for an outcome does not depend upon which measurement that outcome is embedded within, but only upon the mathematical representation of that specific outcome, i.e., its projection operator.Gleason's theorem states that for any such function formula_5, there exists a positive-semidefinite operator formula_2 with unit trace such that

Both the Born rule and the fact that "catalogues of probability" are positive-semidefinite operators of unit trace follow from the assumptions that measurements are represented by orthonormal bases, and that probability assignments are "noncontextual".In order for Gleason's theorem to be applicable, the space on which measurements are defined must be a real or complex Hilbert space, or a quaternionic module.