Let $\mu$ be a finite Borel measure on $\Bbb R$, which is absolutely continuous with respect to the Lebesgue measure $m$. Prove that $x \mapsto \mu(A+x)$ is continuous for every Borel set $A \subseteq \Bbb R$.
Thoughts: $\mid \mu(A+x) -\mu(A-x) \mid = \mid \int_{-\infty}^{A+x} d\mu - \int_{-\infty}^{A+x_0} d\mu \mid = \mid \int_{A+x_0}^{A+x} d\mu \mid =\mid x-x_0 \mid$, hence continuous. I'm not sure whether this is a correct arguement.
2nd thought: sorry, I regarded $\mu$ as a distribution function in probability. Another question, why can we use Radon-Nikodym to write $\mu$ as a convolution?