  The Clauser-Horne-Shimony-Holt (CHSH) inequality is a constraint that local
theories must obey. Quantum Mechanics predicts a violation of this inequality
in certain experimental settings. Treatments of this subject frequently make
simplifying assumptions about the probability spaces available to a local
hidden variable theory, such as assuming the state of the system is a discrete
or absolutely continuous random variable, or assuming that repeated
experimental trials are independent and identically distributed. In this paper,
we do two things: first, show that the CHSH inequality holds even for
completely general state variables in the measure-theoretic setting, and
second, demonstrate how to drop the assumption of independence of subsequent
trials while still being able to perform a hypothesis test that will
distinguish Quantum Mechanics from local theories. The statistical strength of
such a test is computed.
