Bounds for $L_p$-discrepancies of point distributions in compact metric spaces

Upper bounds for the $L_p$-discrepancies of point distributions in compact metric measure spaces for $0<p\le \mathrm{\infty }$ have been established in the paper [6] by Brandolini, Chen, Colzani, Gigante and Travaglini. In the present paper we show that such bounds can be established in a much more general situation under very simple conditions on the volume of metric balls as a function of radii.