  In quantum physics the term `contextual' can be used in more than one way.
One usage, here called `Bell contextual' since the idea goes back to Bell, is
that if $A$, $B$ and $C$ are three quantum observables, with $A$ compatible
(i.e., commuting) with $B$ and also with $C$, whereas $B$ and $C$ are
incompatible, a measurement of $A$ might yield a different result (indicating
that quantum mechanics is contextual) depending upon whether $A$ is measured
along with $B$ (the $\{A,B\}$ context) or with $C$ (the $\{A,C\}$ context). An
analysis of what projective quantum measurements measure shows that quantum
theory is Bell noncontextual: the outcome of a particular $A$ measurement when
$A$ is measured along with $B$ would have been exactly the same if $A$ had,
instead, been measured along with $C$.
  A different definition, here called `globally (non)contextual' refers to
whether or not there is ('noncontextual') or is not ('contextual') a single
joint probability distribution that simultaneously assigns probabilities in a
consistent manner to the outcomes of measurements of a certain collection of
observables, not all of which are compatible. A simple example shows that such
a joint probability distribution can exist even in a situation where the
measurement probabilities cannot refer to properties of a quantum system, and
hence lack physical significance, even though mathematically well-defined. It
is noted that the quantum sample space, a projective decomposition of the
identity, required for interpreting measurements of incompatible properties in
different runs of an experiment using different types of apparatus has a tensor
product structure, a fact sometimes overlooked.
