The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ _ℬs

In the present paper the author uses the function system Γ_ℬsconstructed in Cantor bases to show upper bounds of the extreme and star discrepancy of an arbitrary net in the terms of the trigonometric sum of this net with respect to the functions of this system. The obtained estimations are inequalities of the type of Erdős-Turán-Koksma. These inequalities are very suitable for studying of nets constructed in the same Cantor system.