Continuity of measure-dimension mappings

We study continuity and discontinuity properties of some popular measure-dimension mappings in this work. We give examples to show that no continuity can be guaranteed under general weak, setwise or TV topology on the measure space. However, in some particular circumstances or by assuming some restrictions on the measures, we do have some continuity results. We then apply our continuity results to the case of $t$-conformal measures, to give a sufficient condition on the convergence of the Hausdorff dimensions of the $t$-conformal measures induced from the finite sub-families of an infinite regular CIFS. At last we give some remarks on the density method on deciding the Hausdorff dimensions of measures in our settings.