  Upper bounds for the $L_p$-discrepancies of point distributions in compact
metric measure spaces for $0<p\le\infty$ 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.
