  The use of Bell's theorem in any application or experiment relies on the
assumption of free choice or, more precisely, measurement independence, meaning
that the measurements can be chosen freely. Here, we prove that even in the
simplest Bell test - one involving 2 parties each performing 2 binary-outcome
measurements - an arbitrarily small amount of measurement independence is
sufficient to manifest quantum nonlocality. To this end, we introduce the
notion of measurement dependent locality and show that the corresponding
correlations form a convex polytope. These correlations can thus be
characterized efficiently, e.g., using a finite set of Bell-like inequalities -
an observation that enables the systematic study of quantum nonlocality and
related applications under limited measurement independence.
