A separation theorem with applications to Edgeworth equivalence in some infinite dimensional setting

We provide an Edgeworth Equivalence result for a pure exchange economy with a measure space of agents and a separable ordered Banach space as commodity space. In particular, we first establish a necessary and sufficient condition for the Equivalence to hold, by means of an ad hoc separation theorem, which allows us to extend the classical finite-dimensional proof of this result. Our setting does not require nor transitivity neither completeness of preferences. Finally, using the sufficient part of the previous result and assuming negative transitivity of preferences, we show that the Edgeworth Equivalence holds under a suitable properness ([ro]-uniform weak properness) of preferences.