  Hadwiger's Theorem states that Euclidean-invariant convex-continuous
valuations of definable sets are linear combinations of intrinsic volumes. We
lift this result from sets to data distributions over sets, specifically, to
definable real-valued functions on n-dimensional Euclidean space. This
generalizes intrinsic volumes to (dual pairs) of non-linear valuations on
functions and provides a dual pair of Hadwiger classification theorems.
