the measurable Riemann mapping theorem
The classical Riemann mapping theorem says any simply connected proper plane domain can be mapped conformally to a disc — it produces ANGLE-preserving maps. But what if you want a map that distorts angles in a prescribed way, telling it at every point which direction to stretch and by how much? The measurable Riemann mapping theorem answers exactly this: hand it a field of infinitesimal ellipses (a Beltrami coefficient), even one that varies only measurably, and it builds a quasiconformal map realizing that distortion. It is the engine that turns 'desired distortion data' into 'an actual map.'
Precisely, let mu be a measurable function on the plane (or the sphere) with ess-sup |mu| = k < 1. The Beltrami equation is the PDE f_zbar = mu times f_z, asking for a homeomorphism f whose complex dilatation is the prescribed mu. The theorem (Morrey, then Ahlfors-Bers) asserts: there exists a quasiconformal homeomorphism f of the sphere solving this equation, and f is UNIQUE once normalized (say fixing 0, 1, infinity). Crucially mu need only be MEASURABLE — no continuity, let alone smoothness, is required — and the solution f then depends holomorphically on mu when mu varies in a complex-analytic family (the Ahlfors-Bers parametrized version).
This is the analytic foundation under all of Teichmüller theory. Beltrami coefficients on a surface are precisely the infinitesimal deformations of complex structure, so solving the Beltrami equation is how you move from one Riemann surface to a deformed one; the holomorphic dependence on mu is what gives Teichmüller space its complex-analytic structure. It also undergirds the deformation theory of Kleinian groups (Bers's simultaneous uniformization, sullivan's no-wandering-domains theorem) and complex dynamics (qc surgery). The honest subtlety: 'measurable' is doing enormous work — the same statement with the obvious-looking requirement that mu be continuous would be far weaker and miss the applications, which routinely need merely measurable dilatations supported on fractal sets.
Prescribe mu(z) = 1/2 for |z| < 1 and mu(z) = 0 for |z| > 1 — a Beltrami coefficient that is discontinuous across the unit circle, with |mu| = 1/2 < 1, hence merely measurable. The theorem still produces a quasiconformal homeomorphism f of the sphere with exactly this dilatation: f is 3-quasiconformal (K = (1 + 1/2)/(1 - 1/2) = 3) inside the disc, conformal outside, and the discontinuity in mu causes no obstruction. No classical Riemann-mapping argument reaches such a map; the measurable hypothesis is essential.
Solving a discontinuous Beltrami equation: mu jumps across the circle, yet a unique normalized qc solution still exists.
The power is entirely in the word 'measurable': continuity of mu is NOT required, and that is precisely why the theorem reaches the fractal-supported dilatations of complex dynamics and Kleinian-group deformation. Do not picture it as the classical Riemann mapping theorem with a tweak — it is a global existence-and-uniqueness theorem for a PDE with merely measurable coefficients.