Continuity of the measure of the spectrum for discrete quasiperiodic operators

We study discrete Schrodinger operators (H-alpha,(theta)psi)(n) = psi(n - 1) + psi(n + 1) + f (alphan + theta)psi(n) on l(2)(Z), where f(x) is a real analytic periodic function of period 1. We prove a general theorem relating the measure of the spectrum of H-alpha,(theta) to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of H-alpha,(theta) are positive. For the almost Mathieu operator (f (x) = 2lambdacos 2pix) it follows that the measure of the spectrum is equal to 4\1 - \lambda\\ for all real theta, lambda not equal +/- 1, and all irrational alpha.