  The measurable Riemann mapping theorem proved by Morrey and in some
particular cases by Ahlfors, Lavrentiev and Vekua, says that any measurable
almost complex structure on $\rd$ ($S^2$) with bounded dilatation is
integrable: there is a quasiconformal homeomorphism of $\rd$ ($S^2$) onto $\cc$
($\bc$) transforming the given almost complex structure to the standard one. We
give an elementary proof of this theorem that is done as follows. Firstly we
prove its double-periodic version: each $\ci$ almost complex structures on the
two-torus can be transformed by a diffeomorphism to the standard complex
structure on appropriate complex torus. The proof is based on the homotopy
method for the Beltrami equation on $\td$ with parameter. (As a by-product, we
present a simple proof of the Poincar\'e-K\"obe theorem saying that each
simply-connected Riemann surface is conformally equivalent to either
$\bar{\cc}$, or $\cc$, or the unit disc.) Afterwards the general case is
treated by $\ci$ double-periodic approximation and simple normality arguments
(involving Gr\"otzsch inequality) following the classical scheme.
