  We study continuity and discontinuity of the upper and lower (modified)
box-counting, Hausdorff, packing, (modified) correlation measure-dimension
mappings under the weak, setwise and TV topology on the space of Borel measures
respectively in this work. We give various examples to show that no continuity
can be guaranteed under the weak, setwise or TV topology on the space of Borel
measures for any of these measure-dimension mappings. However, in some
particular circumstances or by assuming some restrictions on the measures, we
do have some (semi-)continuity results for some of these measure-dimension
mappings under the setwise topology. In the end we point out some applications
of our continuity results on deciding the dimensions of measures on ambient
spaces with some dynamical structures.
