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Comprehensive introduction discusses the Möbius transformation, non-Euclidean geometry, elementary transformations, Schwarz's Lemma, transformation of the frontier and closed surfaces, and the general theorem of uniformization. Detailed proofs.

By generalizing the concept of point function to that of a function (""soma"" function) over a Boolean ring, Carathéodory gives in this book an elegant algebraic treatment of measure and integration.