μ non-atomic

μ purely atomic

μ finite

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

· not compact (Theorem 3.7),

· is separable if $\mathcal{E}$ is countably generated (Theorem 3.4).

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

· separable and compact (Theorem 3.8 and Theorem 3.9).

μ infinite

$\left(\tilde{\mathcal{E}}\mathrm{,}\tilde{\mu }\right)$ is:

· not separable (Theorem 3.6),

· not compact (Theorem 3.7).

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

· is separable if μ is outer regular and $\left(E\mathrm{,}\mathcal{E}\right)=\left({ℝ}^d\mathrm{,}\mathfrak{B}\left({ℝ}^d\right)\right)$ .

$\left(\tilde{\mathcal{E}}\mathrm{,}\tilde{\mu }\right)$ is:

· separable and compact if and only if ${\displaystyle \sum _{x\in E_{fin}}}\mu \left(\left\{x\right\}\right)$ is finite and $E_{\infty }$ is finite ((Theorem 3.8 (a) and Theorem 3.9 (a)).

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

· separable if and only if $E_{fin}$ is countable (Theorem 3.8 (b)),

· compact if and only if ${\displaystyle \sum _{x\in E_{fin}}}\mu \left(\left\{x\right\}\right)$ is finite (Theorem 3.9 (b)).