  It is well-known in quantum information theory that a positive operator
valued measure (POVM) is the most general kind of quantum measurement.
Mathematically, a quantum probability is a normalised POVM, namely a function
on certain subsets of a (locally compact and Hausdorff) sample space that
satisfies the formal requirements for a probability measure and whose values
are positive operators acting on a complex Hilbert space. A quantum random
variable is an operator valued function which is measurable with respect to a
quantum probability. In the present work, we study quantum random variables and
generalize several classical limit results to the quantum setting. We prove a
quantum analogue of the Lebesgue dominated convergence theorem and use it to
prove a quantum martingale convergence theorem. This quantum martingale
convergence theorem is of particular interest since it exhibits non-classical
behaviour; even though the limit of the martingale exists and is unique, it is
not explicitly identifiable. However, we provide a partial classification of
the limit through a study of the space of all quantum random variables having
quantum expectation zero.
