  In this paper, we answer a question of Nussbaum, Priyadarshi, and Lunel
[Positive operators and Hausdorff dimension of invariant sets, Trans. Amer.
Math. Soc. 364(2) (2012) 1029-1066.]. We also show that the Hausdorff dimension
and box dimension of the attractor generated by a finite set of contractive
infinitesimal similitudes are the same. Further, we extend many results of
dimension theory to complete metric spaces. In the last part, we fill the gaps
in the proofs of some articles, which are related to the dimension theory, and
hint at some possible improvements in the recent papers.
