I am trying to characterize all measures on $\mathbb{R}$ such that

$$ \sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty, $$

where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, and "$*$" denotes the convolution. This suggests that find whether the measures do not have any growing tails (hence the title of the post).

The space of Radon measures is the dual of continuous functions with compact support. It may have arbitrary large growing tails. For example, $\mu(d x)=e^{|x|}d x$ and $\mu=\sum_{n\in\mathbb{Z}}|n|\: \delta_n$.

Measures which fall in our class include the following examples:

- Absolutely continuous measures with bounded density function, such as the Lebesgue measure.
- $\mu=\sum_{n\in\mathbb{Z}}\delta_n(x).$

Does any one come across this kind of measures? Is this set of measures studied somewhere?

Thanks for any references and remarks!