The paper is concerned with the following (measurable) union problem: let \({\mathcal A}\) be a \(\sigma\)-algebra of subsets of a space X and \(X_ i\in {\mathcal A} (i\in I)\). What conditions on \({\mathcal A}\) and \(X_ i\) guarantee the existence of a subfamily \(X_ i\) (i\(\in J)\) whose union is not measurable, i.e. does not belong to \({\mathcal A}?\) Usually it is assumed that the family \(X_ i (i\in I)\) is point-finite and that \(X_ i\) are small, i.e. belong to some \(\sigma\)-ideal \({\mathcal I}\). Interesting special cases are when \({\mathcal I}\) is the \(\sigma\)-ideal of sets of measure zero for some measure on (X,\({\mathcal A})\) or when X is a topological space and \({\mathcal I}\) is the \(\sigma\)-ideal of meager sets in X.
Satisfactory solutions of the union problem are obtained: (i) for two classes of spaces (X,\({\mathcal A},{\mathcal I})\)- those of perfect and weakly perfect measurable spaces - which are connected to the concept of perfect measures, and (ii) for spaces (X,\({\mathcal A},{\mathcal I})\) which satisfy a Fubini type property and a countability property. This general treatment of the union problem yields most of the results in the literature as special cases.