Uniqueness theorems for simple trigonometric series with application to multiple series

For simple trigonometric series it is shown, in particular, that if the trigonometric series is Riemann summable in measure to an integrable function f and if the Riemann majorant is finite everywhere except possibly on a countable set, then this series is the Fourier series of the function f. Uniqueness theorems for multiple trigonometric series are obtained on the basis of this result.