Measurable Riemann mapping theorem

In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory.

Contrary to its name, it is not a direct generalization of the Riemann mapping theorem, but instead a result concerning quasiconformal mappings and solutions of the Beltrami equation.

The theorem of Ahlfors and Bers states that if μ is a bounded measurable function on C with

, then there is a unique solution f of the Beltrami equation for which f is a quasiconformal homeomorphism of C fixing the points 0, 1 and ∞.

A similar result is true with C replaced by the unit disk D. Their proof used the Beurling transform, a singular integral operator.