LAUSR

Observables on quantum structures can be seen as generalizations of random variables on a measurable space $$(\Omega , \mathcal {A})$$(Ω,A) for the case when $$\mathcal {A}$$A is not necessarily a Boolean algebra. The present paper investigates an extending of the usual pointwise sum of random variables onto the set of bounded observables on a $$\sigma $$σ-distributive lattice effect algebra E. We describe conditions under which this operation, so-called sum $$x+y$$x+y of observables x, y, preserves continuity of spectral resolutions of x, y. We show how the spectrum $$\sigma (x+y)$$σ(x+y) depends on spectra $$\sigma (x)$$σ(x), $$\sigma (y)$$σ(y), and we provide a relation between the meager part $$x_m$$xm and the dense part $$x_d$$xd of an observable x.