Notes on the Regularity of Quasiconformal Homeomorphisms

An analogous theorem holds for quasi-conformal homeomorphisms between subdomains of S, as can easily be seen from the fact that all of the above statements are local in nature. Here a.e. refers to the Lebesgue measure on S. Throughout these notes, all measures will be Borel measures on the appropriate topological space, which will always be some Borel subset of R for some n. ”Almost everywhere” will always refer to the Lebesgue measure on this subset of the appropriate dimension, unless otherwise noted. Lebesgue measure on R will be denoted by mn. When there is no possibility of ambiguity, we will simply write m.