Is there a measure $\mu$ with support $S \subseteq [0,1]$ such that it satisfies:
(i) $S$ has Lebesgue measure zero but is dense on $[0,1]$ with respect to the standard metric;
(ii) $\mu(S)<\infty$; and
(iii) $\forall \epsilon>0$, $\forall a,b \in [0,1]$ such that $b-a\geq \epsilon >0$, $\mu((a,b])\geq k(\epsilon)>0$.
The Cantor distribution fails (i). It is unclear to me that taking the route of the answers here ensures (iii).
The main reason I am asking this is that this possibility is mentioned in Diaconis and Freedman, 1990, p. 1317, but I'm struggling to construct such a measure.
Edit: (iii) should hold $\forall \epsilon>0$. Apologies for the imprecision.