Let \((S,\Sigma,\lambda)\) be a measure space and \(\Sigma_ \lambda\) the associated semimetric space. The main result asserts that under some natural assumptions for an increasing sequence of collections \({\mathcal C}_ n\) of sets in \(\Sigma\) either the union of the collections \({\mathcal C}_ n\) is of first category in \(\Sigma_ \lambda\) or \({\mathcal C}_ n=\Sigma\) for some \(n\).
An application to Nikodým’s uniform boundedness theorem is indicated. Already the special case of the main result that \({\mathcal C}_ n={\mathcal C}\) for all \(n\) is of interest. This is illustrated with several examples of such collections \({\mathcal C}\) arising in a natural way in the theory of vector measures.