## 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)

### MSC:

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