I have the following question concerning an estimate of the total variation norm. Let $\mu$ be a bounded Borel measure on $\mathbb{R}$ and denote by $\mu_t$ the measure defined by $\mu_t(\Omega):=\mu(\Omega+t)$ for $\Omega$ a Borel set and $t$ a real number. I want to show that: $$ \sup_{t\in(0,1]}\frac{1}{t}\|\mu_t-\mu\|<\infty $$ By the definition of the variation norm one obtains: $$ \frac{1}{t}\|\mu_t-\mu\|=\frac{1}{t}\sup_{\pi}\sum_{\Omega\in\pi}{|\mu(\Omega+t)-\mu(\Omega)}| $$ where $\pi$ runs over partitions of $\mathbb{R}$ into countable disjoint measurable subsets. Under what kind of conditions is this uniformly bounded on $(0,1]$? Some continuity property maybe? Another idea is to ask $\mu$ to be differentiable but then I still do not see why the expression should be finite.

Does someone have an idea? Thank you very much in advance.