State Space and Gleason’s Theorem
We begin this chapter by defining the state space of a general logic. We examine the geometric structure of the state space and use it to define the notions of pure states, mixtures of states, and physical properties. Next we define an observable on a logic, allowing us to consider physical experiments whose outcome sets are more general than the finite subsets of ℜ we saw in Chapter 1. We shall be guided by Lemma 1B.10, however, when we define the expected value of an observable as the integral of the identity function on ℜ with respect to a measure determined by a state on the logic.
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