The magnitude of a metric space: from category theory to geometric   measure theory
  Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out to encode many invariants from integral geometry and geometric measure theory, including volume, capacity, dimension, and intrinsic volumes. This paper will give an overview of the theory of magnitude, from its category-theoretic genesis to its connections with these geometric quantities. 
Tom Leinster and Mark W. Meckes
math.MG math.CT