Bartle-Dunford-Schwartz integral versus Bochner, Pettis and Dunford integrals (Q2841070)

Let \((\Omega,\Sigma)\) be a measurable space. Consider a Banach space-valued vector measure \(m:\Sigma \to X\) and a real function \(f\) in the space \(L^1(m)\) (or in the space \(L^1_w(m)\)) of (weakly) integrable functions with respect to \(m\). This interesting paper deals with the relation between the integration of these scalar functions with respect to a vector measure -- the Bartle-Dunford-Schwartz integral -- and the Dunford, Pettis or Bochner integrals of its (vector-valued) distribution function \(m_f\). The distribution function \(m_f\) is the \(X\)-valued function defined as \(m_f(t):=m(\{w \in \Omega:|f(w)|> t\}),\) \( t \geq 0.\) The aim of the authors is to make explicit the fundamental link between the two canonical definitions of vector-valued integration: on the one hand, integration of scalar functions with respect to vector measures; on the other hand, integration of vector-valued functions with respect to scalar measures.NEWLINENEWLINEThe authors show that the Dunford (or Pettis) integrability of \(m_f\) is strongly related to the weak integrability (or the integrability) of \(f\) in the sense of Bartle-Dunford-Schwartz. However, the Bochner integrability of the distribution function \(m_f\) is equivalent to the fact that it belongs to the new space \(L^1(\|m\|)\) of integrable functions with respect to the capacity given by the semivariation \(\|m\|\) of \(m\). This space is defined using the Choquet integral of \(f \) with respect to the semivariation \(\|m\|\) of the measure \(m\). The following results contain the main information of the paper. In these, \(M\) is the \(\sigma\)-algebra of measurable Lebesgue subsets of the half line \([0,\infty)\) and \(\lambda\) is the Lebesgue measure on it, \(D(\lambda,X)\), \(P(\lambda,X)\) and \(B(\lambda,X)\) denote the spaces of \(X\)-valued Dunford integrable functions from the half line, Pettis integrable functions and Bochner integrable functions, respectively.NEWLINENEWLINETheorem 2.4. Let \( f : \Omega\to \mathbb R\) be a measurable function. Then \(f \in L^1_w(m)\) if and only if \(m_g \in D(\lambda,X)\) for all measurable functions \(g\) such that \(|g| \leq |f|\). In this case, we have NEWLINE\[NEWLINE\int_B \langle m_f,x' \rangle\, d\lambda=\int_\Omega \lambda (B \cap [0,|f|])\, d\langle m,x'\rangle, \quad B \in M, \,\, x' \in X'. NEWLINE\]NEWLINE In particular, \(\int_0^\infty \langle m_{f \chi_A},x' \rangle\, d\lambda = \int_A |f|\, d\langle m,x' \rangle\) for all \( A \in \Sigma\) and \(x' \in X'\).NEWLINENEWLINETheorem 2.7. Let \(f:\Omega \to \mathbb R\) be a measurable function. Then \(f \in L^1(m)\) if and only if \(m_g \in P(\lambda,X)\) for all measurable functions \(g\) such that \(|g| \leq |f|\). In this case, we have NEWLINE\[NEWLINE \int_B m_f\, d\lambda= \int_\Omega \lambda(B \cap [0,|f|])\, dm, \quad B\in M. NEWLINE\]NEWLINE In particular, \( \int_0^\infty m_f \chi_A \, d\lambda = \int_A |f|\, dm\) for all \(A \in \Sigma\).NEWLINENEWLINETheorem 3.2. Let \(f:\Omega \to \mathbb R\) be a measurable function.NEWLINENEWLINE(1) If \(f \in L^1(\|m\|)\), then \(m_g \in B(\lambda,X)\) for all measurable functions \(g\) such that \(|g| \leq |f|\).NEWLINENEWLINE(2) If \(X\) is a Banach lattice and \(m\) is a positive vector measure, then \(f\) belongs to \(L^1(\|m\|)\) if and only if \(m_f \in B(\lambda,X)\).NEWLINENEWLINEThe paper is finished by analyzing some inclusions between the spaces involved; for example, the continuous inclusions \(L^\infty(m) + L^1(|m|) \subseteq L^1(\|m\|) \subseteq L^1(m)\) always hold.