The Carathéodory theorem on the construction of a measure is generalized by replacing the outer measure with an approximation of it and generalizing the Carathéodory measurability. The new theorem is applied to obtain dynamically defined measures from constructions of outer measure approximations resulting from sequences of measurement pairs consisting of refining \(\sigma\)-algebras and measures on them which need not be consistent. A particular case when the measurement pairs are given by the action of an invertible map on an initial \(\sigma\)-algebra and a measure on it is also considered.
"On the Carathéodory Approach to the Construction of a Measure." Real Anal. Exchange 42 (2) 345 - 384, 2017. https://doi.org/10.14321/realanalexch.42.2.0345