The sum of observables on a $$\sigma $$ σ -distributive lattice effect algebra

Abstract

Observables on quantum structures can be seen as generalizations of random variables on a measurable space $$(\Omega , \mathcal {A})$$ for the case when $$\mathcal {A}$$ 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 $$\sigma $$ -distributive lattice effect algebra E. We describe conditions under which this operation, so-called sum $$x+y$$ of observables x, y, preserves continuity of spectral resolutions of x, y. We show how the spectrum $$\sigma (x+y)$$ depends on spectra $$\sigma (x)$$ , $$\sigma (y)$$ , and we provide a relation between the meager part $$x_m$$ and the dense part $$x_d$$ of an observable x.