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 its 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 D ρ(A, B) = tr[ρ D(A,B)] and μρ(A)= D ρ(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 f ˆ on H. Continuity and additivity properties of the map f ↦ f ˆ are discussed. It is shown that if f is nonnegative, then f ˆ is a positive operator. A quantum integral is defined by ∫ f d μ ρ = tr ( ρ f ˆ ) . A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.