We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure space leads to the classical space of integrable functions.

The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms where ,  are compact Hausdorff spaces. For a simple case, suppose is metrizable and is such an order isomorphism. By a theorem of Kaplansky, induces a homeomorphism . We prove the existence of a homeomorphism that maps the graph of any onto the graph of . For nonmetrizable spaces the result is similar, although slightly more complicated. Secondly, we let and be compact and extremally disconnected. The theory of the first part extends directly to order isomorphisms . (Here is the space of all continuous functions that are finite on a dense set.) The third part of the paper considers order isomorphisms between arbitrary Archimedean Riesz spaces and . We prove that such a extends uniquely to an order isomorphism between their universal completions. (In the absence of linearity this is not obvious.) It follows, that there exist an extremally disconnected compact Hausdorff space , Riesz isomorphisms of and onto order dense Riesz subspaces of and an order isomorphism such that ( ).

We present a natural way to cover an Archimedean directed ordered vector space by Banach spaces and extend the notion of Bochner integrability to functions with values in . The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map.

We show that D. Lépingle's -inequality extends to the case where we substitute the conditional expectation operators with orthogonal projection operators onto spline spaces and where we can allow that is contained in a suitable spline space . This is done provided the filtration satisfies a certain regularity condition depending on the degree of smoothness of the functions contained in . As a by-product, we also obtain a spline version of - duality under this assumption.