4. An introduction to Connes’ non-commutative integration theory

We are interested in making sense of an integral over a singular space, for example the quotient of the torus T 2 by the lines of irrational slope α / ∈ Q. More generally, we would like to study the quotient space of the global unstable manifolds for a hyperbolic dynamical system. In this chapter we present an “easy” version of Connes’ noncommutative integration theory. We start by studying integer valued functions on a standard measure space, a case in which Connes’ theory and the ordinary integration theory coincide. We then study a simple example of a “singular” measure space, for which the ordinary integration theory is inapplicable.