  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.
