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

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.