The Riemann mapping theorem says that a strict, nonempty open subset of the complex plane is conformally equivalent to the unit disk.

The [measurable Riemann mapping theorem][1] asserts the existence and uniqueness of a quasiconformal homeomorphism $f$ satisfying the Beltrami equation: $$\frac{\partial f}{\partial \overline{z}} = \mu(z)\frac{\partial f}{\partial z} $$ for given $\mu$ with $ \lVert\mu\rVert_{\infty}<1$.
What (if anything) do these two statements have to do with each other? Wikipedia points out in the link above that the latter isn't a direct generalization of the former, although there does seems to be [a proof][2] of Riemann mapping theorem from the measurable RMT.

  [1]: https://en.wikipedia.org/wiki/Measurable_Riemann_mapping_theorem
  [2]: https://en.wikipedia.org/wiki/Beltrami_equation#Smooth_Riemann_mapping_theorem