Submeasurability and integrability of probability functions (Q2706756)

According to the integration theory with respect to a finitely additive measure \(\mu\) developed in the book of \textit{N. Dunford} and \textit{J. T. Schwartz} [``Linear operators''. I (1958; Zbl 0084.10402)] a function \(f\) is integrable iff there is a sequence \((f_n)\) of simple functions converging in measure to \(f\) and being Cauchy in the \(L_1\)-norm; \(\int f d\mu\) is then defined as the limit of the integrals \(\int f_nd\mu\). This concept of an integral is transferred to the case of functions with values in a space \({\mathcal R}^+\) of probability distribution functions. Here is used the concept of a probabilistic submeasure, that is an \({\mathcal R}^+\)-valued monotone function \(\gamma\) defined on an algebra \({\mathcal S}\) of sets satisfying \(\gamma_{A\cup B}(x+ y)\geq T(\gamma_A(x),\gamma_B(y))\) for \(A,B\in{\mathcal S}\) and \(x> 0\), \(y> 0\), where \(T\) is a t-norm.