Conditional expectations operators acting on Riesz spaces are shown to commute with a class of principal band projections. Using the above commutation property, conditional expectation operators on Riesz spaces are shown to be averaging operators. Here the theory of f -algebras is used when defining multiplication on the Riesz spaces. This leads to the extension of these conditional expectation operators to their so-called natural domains, i.e., maximal domains for which the operators are both averaging operators and conditional expectations. The natural domain is in many aspects analogous to L 1 .

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In this work we continue the developments of Kuo et al. (Indag Math 15:435-451, 2004; J Math Anal Appl 303:509-521, 2005) with the construction of the martingale transform or discrete stochastic integral in a Riesz space (measure-free) setting. The discrete stochastic integral is considered both in terms of a weighted sum of differences and via bilinear vector-valued forms. For this, analogues of the spaces L 2 and Mart 2 on Riesz spaces with a conditional expectation operator and a weak order unit are constructed using the f-algebra structure of the universal completion of the Riesz space and properties of the extension of the conditional expectation to its natural domain.