Quantum Measures and Integrals

We show that quantum measures and integrals appear naturally in any L2-Hilbert space H. We begin by defining a decoherence operator D(A, B) and it’s associated q-measure operator µ(A) = D(A, A) on H. We show that these operators have certain positivity, additivity and continuity properties. If ρ is a state on H, then Dρ(A, B) = tr [ρD(A, B)] and µρ(A) = Dρ(A, A) have the usual properties of a decoherence functional and q-measure, respectively. The quantization of a random variable f is defined to be a certain self-adjoint operator fb on H. Continuity and additivity properties of the map f 7→ fb are discussed. It is shown that if f is nonnegative, then fb is a positive operator. A quantum integral is de- fined by R f dµρ = tr(ρfb). A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory