Let A be  such that  $\mu(A) = 1, \lambda(A) = 0$.  There is an open set $O$ with $ A \subset O, \lambda(A) < \epsilon$ Suppose for convenience that $$\liminf\frac{\mu(B(x,r))}{\lambda(B(x,r))}< a $$ on $A$, otherwise replace $A$ with  the set on which it is true. For each x in A  pick $r_x$ so that $$B(x, r_x) \subset O$ \text{ and } $\frac {\mu(B(x,r_{3x}))}{\lambda(B(x,r_{3x}))}< a $$ .  By the Vitale covering lemma there is a disjoint subcollection $J$ with $$A \subset \cup_J B(x, r_{3x})$$ But then $$1 = \mu(a) \le  \sum_J\mu(B(x,r_{3x}) \le a \sum_J \lambda(B(x,r_{3x} )) \le a*3*\epsilon $$ and since $\epsilon$  is at your disposal, this is a contradiction.  I've written this for 1 dimension but I think it applies mutatis mutandis  to all.