  Given a positive operator-valued measure $\nu$ acting on the Borel sets of a
locally compact Hausdorff space $X$, with outcomes in the algebra $\mathcal
B(\mathcal H)$ of all bounded operators on a (possibly infinite-dimensional)
Hilbert space $\mathcal H$, one can consider $\nu$-integrable functions
$X\rightarrow \mathcal B(\mathcal H)$ that are positive quantum random
variables. We define a seminorm on the span of such functions which in the
quotient leads to a Banach space. We consider bistochastic operators acting on
this space and majorization of quantum random variables is then defined with
respect to these operators. As in classical majorization theory, we relate
majorization in this context to an inequality involving all possible convex
functions of a certain type. Unlike the classical setting, continuity and
convergence issues arise throughout the work.
