Given a non-local game G, the n-fold repetition of G can be viewed as a game over the n-th cartesian product of the question and answer sets. When considering the infinite repetition of G, compact topological spaces arise naturally as infinite cartesian products of finite sets. In this talk, we consider the infinite repetition of a finite game as a single infinite game and introduce the notion of a measurable no-signalling correlation by replacing finite sets with second countable, compact, Hausdorff spaces and families of POVMs with operator-valued information channels. We define the measurable counterparts of various classes of no-signalling correlations and focus on the quantum spatial and quantum commuting. To that end, we establish two measurable versions of Stinespring's Dilation Theorem. Finally, we define values of measurable non-local games of local, quantum spatial and quantum commuting type, and show how the asymptotic value of a finite game can be achieved as a special case of the inner value of a measurable game. Based on joint work with Ivan Todorov and Lyudmila Turowska.