Uniformity and covering of standard s-dimensional Hausdorff measures on the line received attention most notably from Fremllin, Shelah and Steprans, (recently) Elekes and Steprans, and others. We calculate uniformity and covering of Hausdorff, packing and Hewitt-Stomberg measures induced by a general gauge on Polish groups and in particular the Cantor set and Baer-Specker group. In particular it turns out that these invariants may differ for the two mentionned groups.