Continuity of measure-dimension mappings with applications to measures originated from IFS.

We study continuity and discontinuity properties of some popular measure-dimension mappings under some topologies on the space of probability measures in this work. We give examples to show that no continuity can be guaranteed under general weak, setwise or TV topology on the space of 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. We then apply our continuity results to concerning measures appearing in several kinds of infinite iterated function systems, namely, CIFS, CGDMS and (families of) PIFS, to show the convergence of the Hausdorff dimensions of the concerning measures induced from the finite sub-systems of these infinite systems. These applications answer a problem of Mauldin-Urba\'nski posed in the last 90s on $t$-conformal measures originally for CIFS positively, and extend a remarkable result of Simon-Solomyak-Urba\'nski from families of finite PIFS to families of infinite PIFS. Finally we indicate more applications of our techniques in some more general circumstances and give some remarks on the relationship between the Hausdorff dimension of measures and their logarithmic density in general settings.