  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\"uders rule and its
generalization (Kraus update rule). Here we investigate the special case of
subsequent measurements where an intermediate measurement is a composition of
two measurements (a or b) and the case where the causal order is not defined (a
and b or b and a). In both cases interference effects can arise. We show that
the associated probability cannot be written univocally, and the distributive
property on its arguments cannot be taken for granted. The two probability
expressions correspond to the Born rule and the classical probability; they are
related to the intrinsic possibility of obtaining definite results for the
intermediate measurement. For indefinite causal order, a causal inequality is
also deduced. The frontier between the two cases is investigated in the
framework of generalized measurements with a toy model, a Mach-Zehnder
interferometer with a movable beam splitter.
