On vector integral inequalities (Q1029958)

Let \(X\) and \(Y\) be complete bornological locally convex spaces and \(m:\Delta\rightarrow L(X,Y)\) be a measure defined on a \(\delta\)-ring with values in the space of continuous linear operators from \(X\) into \(Y\). The authors first present some results obtained in [\textit{J.\,Haluška}, Czech.\ Math.\ J.\ 47, No.\,2, 205--219 (1997; Zbl 0926.46037)] on a generalized Dobrakov integral \(\int f\,dm\) for \(X\)-valued functions. Then they prove some inequalities and equalities, where the semivariation \(\widehat{m}_{U,W}\), the variation \(\text{var}_{U,W}(m,\cdot)\) of \(m\) and the variation \(\text{var}_{U,W}(\int f\,dm,\cdot)\) of the measure \(E\mapsto \int_E f\,dm\) are involved. Here, \(U\) and \(W\) are Banach disks in \(X\) and \(Y\), respectively, and the semivariation and variation are defined with the aid of the Minkowski functionals \(p_U\) and \(p_W\). For example, it is proved that \(\text{var}_{U,W}(\int f\,dm,E)\leq\int_Ep_U(f)d\text{var}_{U,W}(m,E)\) and that \(\widehat{m}_{U,W}(E)=\sup p_W(\int_E f\,dm)\), where the supremum is taken over all integrable function with respect to the Banach spaces \(X_U\), \(Y_W\) and \(p_U(f(t))\leq 1\) for \(t\in E\). Moreover, a ``semivariation'' \(\widehat{m}_{U,W}(g,E)\) for \(X\)-valued function \(g\) and \(E\in\sigma(\Delta)\) is considered.