The space of Radon measures on a complete separable metric space $E$, endowed with the Borel σ-algebra, is denoted by $\mathcal M(E)$, while $\mathcal M_F(E)$ is the subspace of finite measures in $\mathcal M(E)$. The space $\mathcal M_F (E)$ is equipped with the weak topology. Well known fact is that the space $\mathcal M_F (E)$ is a Polish space as well.
Suppose $E$ is a half open interval $[0,H)$ for some $H\in (0,\infty]$. Let $\mathcal M_D[0,H)$ be the subset of measures in $\mathcal M_F ([0,H))$ that can be reprsented as the sum of a finite number of unit Dirac measures in $[0, H)$, that is, measures that take the form $\sum_{i=1}^k \delta_{x_i}$ for some $k\geq1$ and $x_i \in [0,H)$, $i = 1,\ldots,k$. The space $\mathcal M_D [0,H)$ is endowed with the topology of weak convergence.
Question: Is $\mathcal M_D [0,H)$ locally compact?