Two problems on doubling measures (Q1907045)

A measure \(\mu\) on a metric space \(X\) is doubling if \(\mu (B(x,2\rho)) \leq C \mu (B(x,\rho))\) for all open balls \(B(x,2\rho)\), \(B(x,\rho)\) and some \(C>0\). Doubling measures \(\neq 0\) exist on any compact metric space whose metric satisfies a condition of uniform branching, according to a theorem of Vol'berg and Konyagin. If a doubling measure on a compact metric space is continuous, there is a second one, singular with the first (a problem in the work cited). There are compact subsets of \(R^1\) carrying a purely atomic doubling measure and also one which is partly continuous. Besides doubling measures on \(R^1\) there are measures which are ``doubling'' for adjacent dyadic intervals whose union is also dyadic. There is a closed set \(S\) in \([0,1]\) which isn't a null set for dyadic doubling measures; but for almost all real \(t\), \(t+S\) is a null set for that class of measures.