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 D_ρ(A,B)=rmtrsqbrac{ρD(A,B)} and μ_ρ(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 fhat on H. Continuity and additivity properties of the map fmapstofhat are discussed. It is shown that if f is nonnegative, then fhat is a positive operator. A quantum integral is defined by int fdμ_ρ=rmtr \(ρfhat \). A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.