The Measure of the Set of Zeros of the Sum of a Nondegenerate Sine Series with Monotone Coefficients in the Closed Interval $[0,\pi]$

Nonzero sine series with monotone coefficients tending to zero are considered. It is shown that the measure of the set of those zeros of such a series which belong to $[0,\pi]$ cannot exceed $\pi/3$. Moreover, if this value is attained, then almost all zeros belong to the closed interval $[2\pi/3,\pi]$.