On the Carath\'eodory approach to the construction of a measure

TL;DR

This paper generalizes Carathéodory's measure construction by using outer measure approximations and applies it to dynamically defined measures from sequences of measurement pairs, including cases involving invertible maps.

Contribution

It introduces a generalized Carathéodory theorem replacing outer measure with approximations and extends it to dynamic measure constructions from non-consistent measurement sequences.

Findings

Generalized measure construction via outer measure approximations

Application to dynamically defined measures from measurement sequences

Analysis of measures induced by invertible map actions

Abstract

The Carath\'eodory theorem on the construction of a measure is generalized by replacing the outer measure with an approximation of it and generalizing the Carath\'eodory 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.