On uniqueness of multiple trigonometric series

Trigonometric series summable by the Riemann method almost everywhere are considered. In particular, it is proved that if a multiple trigonometric series sums almost everywhere by the Riemann method to an integrable function $f(x)$, and the Riemann majorant of this series satisfies a certain necessary condition, then the series is the Fourier series of the function $f(x)$.