Integration in Riesz spaces with respect to (D)-convergence

Some properties of (D)-convergence (in relation with order convergence) in weakly sigma-distributive and not weakly sigma-distributive Riesz spaces are investigated, in particular dealing with limit-type operators. Some definitions of integral are introduced for Riesz space-valued functions and with respect to finitely additive Riesz space-valued measures, where the Riesz space involved is not necessarily weakly sigma-distributive. Some Vitali-type convergence theorems and Scheffe'-type theorems are proved, with respect to (D)-convergence.