Dilated sections and extremal preimage measures (Q793173)

Let a measurable mapping \(\tau: (Y,{\mathfrak B})\to(X,{\mathfrak A})\) between two measurable spaces be fixed. Define a dilation of \(\tau\) as a triplet \(((S;{\mathfrak S}),\rho,\pi)\) consisting of a measurable space \((S;{\mathfrak S})\), a measurable imbedding \(\rho: Y\to S\) (i.e. \(\rho\) injective and \(\rho\) (\({\mathfrak B})=\rho(Y)\cap {\mathfrak S})\) and a measurable ma\(p\pi: (S;{\mathfrak S})\to(X;{\mathfrak A})\) such that \(\pi {\mathbb{O}}\rho =\tau.\) Given then a fixed measure \(\mu\) on \({\mathfrak A}\) with nonempty set \(M=\{\nu:\nu \quad measure\quad on\quad {\mathfrak B}\quad with\quad \nu {\mathbb{O}}\tau^{-1}=\mu \}\) of preimage measures, it is well-known [cf. \textit{G. A. Edgar}, Ill. J. Math. 20, 630-646 (1976; Zbl 0327.28001)] that image measures \(\nu =\mu {\mathbb{O}}\sigma^{-1}\) to measurable sections \(\sigma\) of \(\tau\) areextremal elements of M. But measurable sections of \(\tau\) exist only under additional hypotheses. The main result of the present paper states however that there is always a dilation \(((S;{\mathfrak S}),\rho,\pi)\) of \(\tau\) characterizing \(ex M\) via sections, i.e. \(\nu\in M\) is extremal in M iff \(\nu {\mathbb{O}}\rho^{-1}=\mu^*{\mathbb{O}}\sigma^{- 1} (=\mu {\mathbb{O}}\sigma^{-1}\), iff \(\tau\) is onto) for some measurable section \(\sigma\) of \(\pi\).