The Bochner and the monotone integrals with respect to a nuclear-valued finitely additive measure (Q2702765)

In this paper the comparison between Bochner and monotone integrals obtained in [\textit{M. C. Isidori, A. Martellotti} and \textit{A. R. Sambucini}, Math. Slovaca 48, No. 3, 253-269 (1998; Zbl 0941.28011)] is extended to the case of complete nuclear spaces. It is used the fact that every complete nuclear space \(E\) is isomorphic to a projective limit of a family \((E_\alpha)_{\alpha \in I}\) of Hilbert spaces. Let \(\Omega \) be an arbitrary nonempty set, \(\Sigma \) a \(\sigma \)-algebra of subsets of \(\Omega \), \(E\) a complete nuclear space. Then \(E\) is the projective limit of a family of Hilbert spaces \((E_\alpha)_{\alpha \in I}\); \(p_\alpha \) denotes the norm on \(E\). Let \(m:\Sigma \to E\) be a boundedly countably additive measure, and for each \(\alpha \in I\) let \(m_\alpha : \Sigma \to E_\alpha \) be the bounded countably additive measure defined by \(m_\alpha (B)=[m(B)]_\alpha \): the semivariation of \(m_\alpha \) is denoted by \(\|m_\alpha \|\). As usual, a measurable function \(f\:\Omega \to \mathbb R\) is \(m\)-integrable if there exists a defining sequence of simple functions, and \(L^1(m)\) denotes the set of all \(m\)-integrable functions. NEWLINENEWLINENEWLINEGiven \(B\in \Sigma \), let \(1_B\) be the indicator function associated with \(B\). Given a measurable function \(f:\Omega \to [0,+\infty)\), and \(B\in \Sigma \) let \(\varphi^B\), \(\varphi^B_\alpha \), \(\alpha \in I\), be defined as \(\varphi^B(t)=m({f\cdot 1_B>t})\), \(\varphi^B_\alpha (t)=m_\alpha ({f\cdot 1_B>t})\). Then \(f\) is said to be \((\sphat )\)-integrable if for each \(B\in \Sigma \), \(\varphi^B\) is integrable by seminorm on \(([0,+\infty),\mathcal B, \mu)\) (where \(\mathcal B\) is the Borel \(\sigma \)-algebra and \(\mu \) is the Lebesgue measure) and for every \(\alpha \in I\), \(F_\alpha \in L^1([0,+\infty), \mathcal B, \mu)\) exists such that \(p_\alpha (\varphi^B_\alpha (t))\leq F_\alpha (t)\), for all \(t\in [0,\infty)\). NEWLINENEWLINENEWLINEIf \(f\) is real-valued, then \((\sphat )\)-integrability is given in terms of \(f^+\) and \(f^-\). \(\widehat L^1 (m)\) denotes the set of all \((\sphat )\)-integrable functions. NEWLINENEWLINENEWLINEIn the paper it is proven that every measurable real-valued function \(f\) is \(m\)-integrable if and only if it is \(m_\alpha \)-integrable for every \(\alpha \in I\), and is \((\sphat )\)-integrable with respect to \(m\) if and only if \(f\) is \((\sphat )\)-integrable with respect to \(m_\alpha \) for each \(\alpha \in I\). The result relative to \((\sphat )\)-integrability is obtained even for s-bounded finitely additive measures \(m\). The last part of the paper deals with integration with respect to orthogonally scattered dilations, and is an extension of the results obtained in [\textit{M. C. Isidori, A. Martellotti} and \textit{A. R. Sambucini} (loc. cit.)].