Lineability and spaceability on vector-measure spaces (Q2866763)

A subset \(M\) of an infinite-dimensional Banach space \(E\) is said to be lineable if \(M \cup \{0 \}\) contains an infinite-dimensional vector subspace of \(E\), and spaceable if \(M \cup \{0 \}\) contains an infinite-dimensional closed vector subspace of \(E\). Let \(I= [0, 1]\), \(\mathcal{B}\) is the class of all Borel subsets of \(I\), and \(\lambda\) the Lebesgue measure on \(I\). \(X\) is an infinite-dimensional Banach space and \(ca(\mathcal{B}, \lambda, X)\) is the Banach space of all countably additve measures on \(\mathcal{B}\) which are absolutely continuous wrt \(\lambda\) with norm \(= \sup_{A \in \mathcal{B}} \| \mu(A) \|_{X} \); for a \(\mu \in ca(\mathcal{B}, \lambda, X), \; |\mu|: \mathcal{B} \to [0, \infty]\) denotes its total variation measure. A measure \(\mu \in ca(\mathcal{B}, \lambda, X)\) is said to be injectable if, for any \(f, g \in L_{\infty}(\lambda)\), \(\int f d \mu =\int g d \mu\) implies \(f=g \; a.e (\lambda)\).NEWLINENEWLINEIn this paper, the authors extend some known results about the lineability and spaceability of some subsets of \(ca(\mathcal{B}, \lambda, X)\). The major results are:NEWLINENEWLINEI. The set of measures \(\mu\) in \( ca(\mathcal{B}, \lambda, X)\), with \(\mu\) having a relatively compact range and \(|\mu|(A)= \infty\) for every non-null set \(A\), is lineable in \(ca(\mathcal{B}, \lambda, X)\).NEWLINENEWLINEII. Let \(M_{\sigma}\) be the subset of \(ca(\mathcal{B}, \lambda, X)\) consisting of measures whose total variation measures are \(\sigma\)-finite. Then \(ca(\mathcal{B}, \lambda, X) \setminus M_{\sigma}\) is spaceable in \(ca(\mathcal{B}, \lambda, X)\).NEWLINENEWLINEIII. The set of injective measures is spaceable in \(ca(\mathcal{B}, \lambda, X)\).