Conditional expectations on Riesz spaces

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 L¹.