Bochner integrals in ordered vector spaces
  We present a natural way to cover an Archimedean directed ordered vector space $E$ by Banach spaces and extend the notion of Bochner integrability to functions with values in $E$. The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map. 
Arnoud van Rooij and Willem van Zuijlen
math.FA