Vector measures and integration in non-complete spaces
Given a locally convex Hausdorff space \(X\) (briefly, lcHs) and a \(\sigma\)-additive measure \(m: \Sigma\to X\), where \(\Sigma\) is a \(\sigma\)- algebra, it is possible to construct an associated \(L^ 1\)-space. An important property is that \(| f|\in L^ 1(m)\) whenever \(f\in L^ 1 (m)\). So, it is natural to consider the theory in a complex lcHs \(X\) which need not be sequentially complete. However, in this generality Murphy's law takes over: the classical convergence theorems may fail, bounded \(\Sigma\)-measurable functions may not be integrable, an element \(f\in L^ 1 (m)\) may have the property that \(\overline {f}\) for \(| f|\) is not integrable. Let \(X[ m]\) be the sequential closure of the space spanned by the range \(m(\Sigma)\) of \(m\). Then, the sequential completeness of \(X[ m]\) is a sufficient condition ensuring that \(L^ 1 (m)\) once again has all the familiar properties expected of it. Under a certain closedness condition on \(m\) it even guarantees that \(L^ 1 (m)\) is complete. Finally, there are many examples of measures \(m\) in non- sequentially complete spaces \(X\) for which \(X[ m]\) is sequentially complete. Connecting properties of the space \(X[ m]\) with those of \(L^ 1 (m)\) form the basis of this paper.
scientific article; zbMATH DE number 687030