Prove that if $\lambda$ is absolutely continuous with respect to the Lebesgue measure $L^n$ and $\lim_{r\to 0} \frac{\lambda(B(x, r))}{r^{n}} = 0$ for every $ x ∈ \mathbb{R}^n$ then $\lambda = 0$.
If I could show that $L^n=0$ then $\lambda=0$ from being absolutely continuous w.r.t $L^n$. I am thinking that the Radon-Nikodym theorem would be helpful, as $\lambda<<L^n$, then we can find $h\geq 0$ such that $\lambda(\mathbb{R}^n)=\int_{\mathbb{R}^n} h $ $ dL^n$. However, I can't seem to use it. Any ideas? Try not to give full solutions, thank you.