Conditional probability and interferences in generalized measurements

In the context of generalized measurement theory, the Gleason-Busch theorem assures the unique form of the associated probability function. Recently, in Flatt et al. Phys. Rev. A 96, 062125 (2017), the case of subsequent measurements has been treated, with the derivation of the L{u}ders rule and its generalization (Krauss update rule). Here we investigate the special case of subsequent measurements where an intermediate measurement is a composition of two measurements (a or b) with possible interference effects. In this case, the associated probability cannot be written univocally, and the distributive property on its arguments cannot be taken for granted. Different probability expressions are related to the intrinsic possibility of obtaining definite results for the intermediate measurement. The frontier between the two cases is investigated in the framework of generalized measurements with a toy model, a Mach-Zehnder interferometer with movable beam splitter.