In the lecture we looked at the theorem of Carathéodory. As a motivation to it the following is stated: If $E$ is a generating system for a $\sigma$-algebra, it is in general impossible to extend a measure $\mu: E \rightarrow \mathbb{R}$ to a measure $\overline{\mu}: \sigma(E) \rightarrow \mathbb{R}$.
But no example was given. I've also found this statement online several times, but no one gave an example. Is it something obvious? This has been written more explicit in our homework:
a) Find a finite set $\Omega$, $E \subset \mathcal{P}(\Omega)$, and a map $\mu: E \rightarrow [0,\infty)$ which fulfills the following:
i) $\emptyset \in E$
ii) $\mu(\emptyset)=0$ and $\mu$ is $\sigma-$additive for disjoint unions and otherwise sub-additive.
iii) No extension exists from $E$ to $\sigma(E)$.
I've tried to think about such a generating system $E$, but I can't find one that shows the above since I don't understand how one would show that there can't be such an extended measure.
b) If $E$ is additionally to the above closed under finite intersections, can one still find such an example?
I don't believe that's true, but I fail to see why. I'm assuming that "closed under finite intersections" will lead to the fact, that for all $A,B \in E$ there are $C_1, \dots, C_n \in E$ such that $B\backslash A=C_1 \cup \dots \cup C_n$. Thus, $E$ becomes a semi-ring (so we can use Carathéodory), but I don't know how to proof that. Or is that even the right way?
Can someone help me here? Thanks in advance.