A celebrated theorem of Ornstein and Weiss states that any pair of free ergodic probability preserving actions of two amenable groups are orbit equivalent. In this course I will present a way to "quantity" orbit equivalence by considering certain integrability conditions on the OE-coupling. First we will prove that the Folner profile is partially preserved under such a condition. This will imply that the couplings in Ornstein-Weiss theorem cannot satisfy any uniform integrability condition, as weak as it might be. Conversely, we will present a new method for constructing explicit OE between amenable groups, which comes with a control on the integrability. 

Several examples will be discussed.