Could you please give me some hints on (Exercise 1.7.21) of "A Course in Metric Geometry" by Burago, Burago, Ivanov.
We have a compact space $X$, which can be written as $X=\bigcup_{i=1}^n X_i$ (disjoint union), where $X_i=F_i(X)$ and $F_i$ are dilations with Lipschitz constant $c_i$. Then show that the Hausdorff dimension $d$ of $X$ satisfies $\sum_{i=1}^n c_i^d = 1$.
Let $\mu_d$ denote the $d$-dimensional Hausdorff measure, then thats how far I got:
Thanks to the corrections below I am able to show it in the case where $0<\mu_d(X)<\infty$ but I still do not know how to approach the two boundary cases. Any hints are appreciated! Thanks!