$\left(\tilde{\mathcal{E}}\mathrm{,}{d}_{\mu }\right)$

$\left({\tilde{\mathcal{E}}}_{fin}\mathrm{,}{d}_{\mu }\right)$

Separable

for $\mathcal{E}$ countably generated if and only if μ is finite (Theorem 3.4 and Theorem 3.7).

· for ${\tilde{\mathcal{E}}}_{fin}={\widetilde{\mathfrak{B}\left({ℝ}^d\right)}}_{fin}$ , if μ is outer regular (Theorem 3.3),

· for μ purely atomic measure where all the atoms are singletons, if and only if the set of atoms with finite measure $E_{fin}$ is countable (Theorem 3.8).

Compact

if and only if μ is purely atomic with all atoms singletons and ${\displaystyle \sum _{x\in E_{fin}}}\mu \left(\left\{x\right\}\right)<\infty$ and and $E_{\infty }$ is finite (Theorem 3.7 and Theorem 3.8 (a)).

for μ is purely atomic measure where all atoms are singletons if and only if ${\displaystyle \sum _{x\in E_{fin}}}\mu \left(\left\{x\right\}\right)<\infty$ (Theorem 3.9).