Density points outside of a non-measurable set

Question

Let $\lambda$ be a Radon outer measure over $R^n$ (that is: all Borel sets are $\lambda$-measurable in Caratheodory sense, for every set $C$ there exists a Borel set $B$, $C \subset B$ such that $\lambda(C) = \lambda(B)$ and $\lambda$ is finite on compact sets) and let $A \subset R^n$.
Show that if $\lim \limits_{r \to 0}$ $\frac{\lambda(A \cap B(x,r))}{\lambda(B(x,r))} = 0$ for $\lambda$-almost every $x \in R^n\setminus A$ then $A$ is a $\lambda$ measurable set. $B(x,r)$ is an open ball centered at $x$ with radius $r$.
I tried using Lebesgue differentiation theorem for some indicator functions and outer regularity of the measure, but it doesn't seeem to work. Maybe some covering theorem could work?
I'm stuck, please help.