Lineability in subsets of measure and function spaces
A subset \(M\) of a function space is called lineable (or \(c\)-lineable) if \(M\cup \{0\}\) contains an infinite dimensional linear subspace (or, respectively, a linear subspace of dimension \(c\)). The main results are the following. Let \(1\leq p <q\). Then \(L^p[0,1]\setminus L^q[0,1]\) is \(c\)-lineable. If \(I\) is an unbounded interval, then \(L^q(I)\setminus L^p(I)\) is \(c\)-lineable. \(l_q\setminus l_p\) is \(c\)-lineable. Let \(\lambda\) be the Lebesgue measure on the Borel sets in \([0,1]\) and \(X\) an infinite dimensional Banach space. Then the set of injective measures is lineable in \(ca(\lambda,X)\). The set of measures whose range is neither closed nor convex is lineable in \(ca(\lambda,X)\). If \(\overline{\lambda}\) is the Lebesgue measure on the measurable subsets of \([0,1]\) and \(1\leq p<\infty\), then the set of \(l_p\)-valued measures with relatively compact range and such that their total variation is \(\infty\) on every non-null set is lineable in \(cca(\overline{\lambda},l_p)\).
scientific article; zbMATH DE number 5275766
scientific article