Convergence of Riemann sums for functions which can be represented by trigonometric series with coefficients forming a monotonic sequence

The following theorem is proved. If $$f(x) = frac{{alpha _0 }}{2} + sumnolimits_k^infty alpha _k cos 2pi kx + b_k sin 2pi kx,$$ wherea _k ↓ 0 and b_k ↓ 0, then $$mathop {lim }limits_{n to infty } frac{1}{n}sumnolimits_{s = 0}^{n - 1} {fleft( {x + frac{s}{n}} right) = frac{{alpha _0 }}{2}} $$ on (0, 1) in the sense of convergence in measure. If in additionf(x) ε L² (0, 1), then this relation holds for almost all x.