On the Carathéodory approach to the construction of a measure. (English) Zbl 1392.28005

From the introduction: “We generalize the Carathéodory measurability, prove a generalization of the Carathéodory theorem for outer measure approximations and develop a general theory for such constructions. It naturally extends the classical measure theory and can be called dynamical measure theory.” The author defines a “dynamically defined measure” as follows: Let \(\mathcal{A}_0\subset\mathcal{A}_{-1}\subset\mathcal{A}_{-2}\dots\) be a sequence of \(\sigma\)-algebras on \(X\) and \(\phi_i:\mathcal{A}_{i}\rightarrow [0,+\infty]\) a (\(\sigma\)-additive) measure on \(\mathcal{A}_{i}\). Let \(\Phi_i(Q):=\inf\{\sum_{m\leq i} \phi(A_m): A_m\in \mathcal{A}_m\,,\, Q\subset \cup_{m\leq i}A_m\} \) and \(\bar{\Phi}(Q):=\lim_{i\rightarrow -\infty}\Phi_i(Q)\) for \(Q\subset X\). Then \(\bar{\Phi}\) is an outer measure. The restriction of \(\bar{\Phi}\) on the \(\sigma\)-algebra \(\mathcal{A}_{\bar{\Phi}}\) of Carathéodory \(\bar{\Phi}\)-measurable sets is called a dynamically defined measure. As shown, \(\mathcal{A}_i\subset\mathcal{A}_{\bar{\Phi}}\) for \(i\leq 0\).
Reviewer: Hans Weber (Udine)


28A12 Contents, measures, outer measures, capacities
28A99 Classical measure theory
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