I have been really puzzled as to why he would require the $\sigma$-algebra $\mathfrak{M}$. What significances does $E\cap K\in \mathfrak{M}_F$ for all compact sets, have?
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$\begingroup$Consider the measure $$\delta_{\mathbb Z}:= \sum_{z \in \mathbb Z} \delta_z$$
Then, $\mathbb R \notin \mathfrak{M}_F$, which is a problem. But, you do have $\mathbb R \in \mathfrak{M}$.
Also, most importantly, $\mathfrak{M}_F$ is not a $\sigma$-algebra in this case, as $[n,n+1] \in \mathfrak{M}_F$ but $\bigcup_n [n,n+1] \notin \mathfrak{M}_F$. In general, $\mathfrak{M}_F$ is a $\sigma$-algebra only if the measure is finite (in which case you have $\mathfrak{M}_F=\mathfrak{M}$).
$\mathfrak{M}$ is always a $\sigma$-algebra, and it is the right object,but it is hard to define it directly, without referring first to $\mathfrak{M}_F$.
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