  A quantum probability measure is a function on a sigma-algebra of subsets of
a (locally compact and Hausdorff) sample space that satisfies the formal
requirements for a measure, but whose values are positive operators acting on a
complex Hilbert space, and a quantum random variable is a measurable operator
valued function. Although quantum probability measures and random variables are
used extensively in quantum mechanics, some of the fundamental probabilistic
features of these structures remain to be determined. In this paper we take a
step toward a better mathematical understanding of quantum random variables and
quantum probability measures by introducing a quantum analogue for the expected
value of a quantum random variable relative to a quantum probability measure.
In so doing we are led to theorems for a change of quantum measure and a change
of quantum variables. We also introduce a quantum conditional expectation which
results in quantum versions of some standard identities for Radon-Nikodym
derivatives. This allows us to formulate and prove a quantum analogue of Bayes'
rule.
