An AndA′-Douglas type theorem in Riesz spaces with a conditional expectation

In this paper we formulate and prove analogues of the Hahn-Jordan decomposition and an AndA '-Douglas-Radon-NikodA1/2m theorem in Dedekind complete Riesz spaces with a weak order unit, in the presence of a Riesz space conditional expectation operator. As a consequence we can characterize those subspaces of the Riesz space which are ranges of conditional expectation operators commuting with the given conditional expectation operators and which have a larger range space. This provides the first step towards a formulation of Markov processes on Riesz spaces.