Martingales and some integrability criterions for multiple trigonometric series

Applying some martingale properties of the sequences of cubic partial sums of series in the Haar multiple system, some integrability criterions for multiple trigonometric series are established. Some of the proved theorems improve and extend to the multiple trigonometric series the classical theorems on integrability of one-dimensional trigonometric series, the proofs of which are based on Helly??s principal of choice.