The Fatou property in \(p\)-convex Banach lattices (Q864690)

Let \((\Omega, \Sigma)\) be a measurable space, \(X\) a Banach space and \(X^\ast\) be its dual. Let \(B_{X^\ast}\) be the closed unit ball of \(X^\ast\). Let \(\vartheta:\Sigma\to X\) be a \(\sigma\)-additive vector measure. For \(x^\ast\in X^\ast\), let \(x^\ast\vartheta\) be the real-valued measure \(A\mapsto\langle x^\ast,\vartheta (A)\rangle\) and \(| x^\ast \vartheta| \) its variation measure. A measurable function \(f:\Omega\rightarrow\mathbb{R}\) is integrable with respect to \(\vartheta\) if (i) \(f\) is \(x^\ast\vartheta\) integrable for every \(x^\ast\in X^\ast\), and (ii) for every \(A\in\Sigma\), there exists \(\int_A f\,d\vartheta\) in \(X\) such that \[ \langle x^\ast,\int_A f \,d \vartheta\rangle = \int_A f \,dx^\ast\vartheta\text{ for each }x^\ast\in X^\ast. \] The space \(L^1 (\vartheta)\) of all \(\vartheta\)-integrable functions is a Banach space for the norm \[ \| f\|_1 = \sup_{x\in B_{X^\ast}}\int_\Omega | f|\,d| x^\ast\vartheta|. \] Functions satisfying property (i) above are said to be \(\vartheta\)-scalarly integrable and are denoted by \(L_w^1 (\vartheta)\). \(L_w^1 (\vartheta)\) is a Banach lattice for the \(\vartheta\)-a.e.\ order. For \(1 \leq p < \infty\), let \(L^p (\vartheta)\) denote the space of all functions \(f:\Omega\to\mathbb{R}\) such that \(| f|^p\in L^1(\vartheta)\), equipped with the norm \[ \| f \|_p = \sup_{x^\ast \in X^\ast} (\int_\Omega | f| ^p \,d| x^\ast\vartheta|)^{1/p}.\tag{1} \] \(L^p(\vartheta)\) is a Banach lattice with order continuous norm. The space \(L_w^p(\vartheta)\) is the set of all functions \(f:\Omega\rightarrow\mathbb{R}\) with \(| f|^p\in L_w^{1}(\vartheta)\). It is a Banach lattice for the norm in (1). \(L_w^p(\vartheta)\) is a \(p\)-convex Banach lattice and the elements in \(L_w^p(\vartheta)\) with \(\sigma\)-order continuous norm can be identified with \(L^p(\vartheta)\). \(L_w^p(\vartheta)\) has the \(\sigma\)-Fatou property, \(\chi_\Omega\) is a weak unit in \(L_w^p(\vartheta)\) which belongs to \(L^p(\vartheta)\). In the main theorem of the paper, the authors show that these properties of \(L_w^p (\vartheta)\) characterize a large class of Banach lattices. Theorem. Let \(1\leq p <\infty\) and \(E\) be any \(p\)-convex Banach lattice with the \(\sigma\)-Fatou property and possessing a weak unit which belongs to the \(\sigma\)-order continuous part of \(E\). Then there exists a vector measure \(\vartheta\) such that \(E\) is Banach lattice isomorphic to \(L_w^p(\vartheta)\).