Image measures and the so-called image measure catastrophe (Q1380147)

Let \(X\) and \(Y\) be nonempty sets, let \({\mathfrak A}\) be a \(\sigma\)-algebra of subsets of \(X\), and let \(\alpha:{\mathfrak A}\to [0,\infty]\) be a measure. For a map \(H: X\to Y\) denote by \(H[{\mathfrak A}]\) the family of all \(B\subset Y\) with \(H^{-1}(B)\) in \({\mathfrak A}\) and set \(H[\alpha](B)= \alpha(H^{-1}(B))\) for \(B\) in \(H[{\mathfrak A}]\). Clearly, \(H[{\mathfrak A}]\) is a \(\sigma\)-algebra of subsets of \(Y\) and \(H[\alpha]\) is a measure, called the image of \(\alpha\) under \(H\). Let further \({\mathfrak B}\) be a sub-\(\sigma\)-algebra of \(H[{\mathfrak A}]\) and set \(\beta= H[\alpha]|{\mathfrak B}\). Obviously, \(H(X)\) need not belong to the (outer) Carathéodory class \(\text{Meas}(\beta)\) of \(\beta\), in general. The fact that this can be so ``even for topological spaces and Borel \(\sigma\)-algebras'' was called the image measure catastrophe by \textit{L. Schwartz} [``Radon measures on arbitrary topological spaces and cylindrical measures'' (1973; Zbl 0298.28001), p. 30]. The author presents inner regularity conditions on \(\alpha\) which are necessary and sufficient in order that \(H(X)\in\text{Meas}(\beta)\), resp. \(H[{\mathfrak A}]\subset \text{Meas}(\beta)\). He is also concerned with the inner regularity of \(H[\alpha]\) in the case where \(\alpha\) is inner regular with respect to a sublattice of \({\mathfrak A}\). In a topological situation this yields a theorem on the image of a Radon measure under a Lusin measurable map related to some results of \textit{L. Schwartz} [op. cit., Section I.5]. Moreover, the paper contains a discussion of the essential outer measure generated by \(\alpha\) and \(\alpha\)-integrable and \(\alpha\)-essentially integrable sets as defined by \textit{L. Schwartz} [op. cit., Section I.1]. The details are mostly rather technical. The author uses extensively the notation, terminology and results of his recent book [``Measure and integration. An advanced course in basic procedures and applications'' (1997; Zbl 0887.28001)].