  We introduce the notion of scale to generalize and compare different
invariants of metric spaces and their measures. Several versions of scales are
introduced such as Hausdorff, packing, box, local and quantization. They
moreover are defined for different growth, allowing in particular a refined
study of infinite dimensional spaces. We prove general theorems comparing the
different versions of scales. They are applied to describe geometries of
ergodic decompositions, of the Wiener measure and of functional spaces. The
first application solves a problem of Berger on the notions of emergence
(2020); the second lies in the geometry of the Wiener measure and extends the
work of Dereich-Lifshits (2005); the last refines Kolmogorov-Tikhomirov (1958)
study on functional spaces.
