  We present a mathematical framework based on quantum interval-valued
probability measures to study the effect of experimental imperfections and
finite precision measurements on defining aspects of quantum mechanics such as
contextuality and the Born rule. While foundational results such as the
Kochen-Specker and Gleason theorems are valid in the context of infinite
precision, they fail to hold in general in a world with limited resources. Here
we employ an interval-valued framework to establish bounds on the validity of
those theorems in realistic experimental environments. In this way, not only
can we quantify the idea of finite-precision measurement within our theory, but
we can also suggest a possible resolution of the Meyer-Mermin debate on the
impact of finite-precision measurement on the Kochen-Specker theorem.
