Some Baire category results for measures with `big' ranges (Q1280009)

The paper gives a far reaching generalization of the following result due to I. Labuda and the first author: For every sequence \((m_n)\) of atomless scalar-valued measures defined on a \(\sigma\)-algebra \(\Sigma\) there exists a set \(A\in\Sigma\) with \(m_n(A)\neq 0\) for all \(n\). For this, the authors introduce a class of subsets \({\mathcal H}(X)\) (resp. \({\mathcal H}_\sigma(X)\)) of \(X\), called `small sets' (resp. `\(\sigma\)-small sets'), which satisfies some mild axioms where \(X\) is a topological Abelian group or vector space. Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of a set \(S\) and \(\eta\) an order continuous submeasure on \(\Sigma\). Under the Fréchet-Nikodým topology induced by \(\eta\), \((\Sigma,\eta)\) is a complete pseudometric space and hence the Baire category theorem holds. One of the principal theorems is the following: Let \(m_n:\Sigma\to X\) be measures dominated by an order continuous submeasure \(\eta\) on \(\Sigma\) and let \(\{m_n(E):E\in \Sigma\}\not\in{\mathcal H}(X)\), \(n= 1,2,\dots\)\ . Then, for every sequence \((L_n)\) of sets from \(\overline{\mathcal H}_\sigma(X)\) (resp. \({\mathcal H}_\sigma(X)\)), the class \({\mathcal A}= \{A\in\Sigma: m_n(A)\not\in L_n\) for each \(n\}\) is dense and \(G_\delta\) (resp. residual) in \((\Sigma, \eta)\).