Let $\mathscr M(\mathbb R)$ be the Banach space of complex-valued Radon measures on $\mathbb R$, and let $\pi$ be the action of $\mathbb R$ on $\mathscr M(\mathbb R)$. Let $\mathscr A$ denote a subset of $\mathscr M(\mathbb R)$ such that $\pi_x(\mu)$ is continuous with respect to $x$ for every $\mu \in \mathscr A$.
My question is what does $\mathscr A$ looks like?