High Quality Content by WIKIPEDIA articles! In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem states that the space of "measures" (see below) defined on finite unions of compact convex sets in Rn consists of one "measure" that is "homogeneous of degree k" for each k = 0, 1, 2, ..., n, and linear combinations of those "measures". Here "measure" means a real-valued function m that is invariant under rigid motions (combinations of rotations and translations), finitely additive (if A and B are finite unions of compact convex sets then m(A ? B) = m(A) + m(B) ? m(A ? B), and m(?) = 0), and convex-continuous (its restriction to convex sets is continuous with respect to the Hausdorff metric). The countable additivity condition that is usually a part of the definition of measure is not required here.

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