Conditional expectations in generalized probability theory

Expectations have been considered as dual objects of instruments in several papers on generalized probability theory and quantum theory. Here, we relate a generalized conditional (GC) expectation to a given instrument and a given state by two requirements, which are analogous to the axioms by which the classical conditional expectation is related to a given sub-σ-algebra and a given probability measure. In examples we illustrate the close similarity of GC expectations with classical conditional expectations. Eventually, we introduce a rich class of quantum stochastic processes, which are Markovian.