 The probability of any outcome of a measurement upon a quantum system must be a real number between 0 and 1 inclusive, and in order to be consistent, for any individual measurement the probabilities of the different possible outcomes must add up to 1.Gleason's theorem shows that any function that assigns probabilities to measurement outcomes, as identified by projection operators, must be expressible in terms of a density operator and the Born rule.This gives not only the rule for calculating probabilities, but also determines the set of possible quantum states.