Operators represented by conditional expectations and random measures (Q850581)

On standard measure spaces (spaces which are isomorphic to Borel subsets of complete separable metric spaces), every order bounded order continuous linear operator acting between ideals of measurable functions is generated by a random measure. The authors prove an analogue of this result for arbitrary \(\sigma\)-finite measure spaces. This then leads to the main result of the paper. Theorem. Every order continuous linear map between ideals of almost everywhere finite measurable functions on a \(\sigma\)-finite measure space is multiplication conditional expectation representable. These considerations shed further light on the structure of order continuous operators and show that multiplication operators, Riesz homomorphisms and conditional expectations constitute the building blocks of every order continuous operator.