Quantum measures and integrals

We show that quantum measures and integrals appear naturally in any $L_2$-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 $D_ρ(A,B)=\rmtr\sqbrac{ρD(A,B)}$ 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 $f\mapsto\fhat$ are discussed. It is shown that if $f$ is nonnegative, then $\fhat$ is a positive operator. A quantum integral is defined by $\int fdμ_ρ=\rmtr (ρ\fhat\,)$. A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.

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