The measure derivative has unbounded variation
Let \((X, \Sigma, \mu)\) be a measure space, where \(X\) is locally convex, \(\Sigma\) contains all the cylindrical sets in \(X\) and \(\mu\) is a finite measure. Let \(D(\mu)\) be the set of all vectors in whose direction \(\mu\) is differentiable. Let \(W\) be a linear subspace of \(D(\mu)\), normed by \(h\mapsto |d_n \mu|\). Define a \(W^*\)-valued vector measure \(\mu'\) by \(\langle \mu' (A), h\rangle= d_h \mu(A)\), \(A\in \Sigma\). The author shows that \(\mu'\) has bounded variation if and only if \(\dim W<\infty\). This result was known under the restriction that \(W= D(\mu)\) is a Hilbert space.
scientific article; zbMATH DE number 640208