  A characterization of noncontextual models which fall within the ambit of
Fine's theorem is provided. In particular, the equivalence between the
existence of three notions is made explicit: a joint probability distribution
over the outcomes of all the measurements considered, a
measurement-noncontextual and outcome-deterministic (or KS-noncontextual, where
'KS' stands for 'Kochen-Specker') model for these measurements, and a
measurement-noncontextual and factorizable model for them. A KS-inequality,
therefore, is implied by each of these three notions. Following this
characterization of noncontextual models that fall within the ambit of Fine's
theorem, non-factorizable noncontextual models which lie outside the domain of
Fine's theorem are considered. While outcome determinism for projective (sharp)
measurements in quantum theory can be shown to follow from the assumption of
preparation noncontextuality, such a justification is not available for
nonprojective (unsharp) measurements which ought to admit
outcome-indeterministic response functions. The Liang-Spekkens-Wiseman (LSW)
inequality is cited as an example of a noncontextuality inequality that should
hold in any noncontextual model of quantum theory without assuming
factorizability. Three other noncontextuality inequalities, which turn out to
be equivalent to the LSW inequality under relabellings of measurement outcomes,
are derived for Specker's scenario. The polytope of correlations admissible in
this scenario, given the no-disturbance condition, is characterized.
