Let $X$ be a set, with $E_1, E_2,...$ subsets, we define $A$={$x \in X : x \in E_n $ for infinite $n$}. I want to prove something like the Borel-Cantelli for the outer measure $\mu^*$:
a) If $\mu^* (X) < \infty$, then $\mu^* (A) = 0$
b) Also, if we define that $\mu$ is the Caratheodory's restriction for $\mu^*$ (i.e. the restriction of $\mu^*$ to the $\sigma$-algebra $\text M$ of all the subsets $D$ of X such that $\mu^*(B)=\mu^*(B \cap D) + \mu^*(B \cap D^c)$), $\mu (X) < \infty$, $inf \ \mu (E_n) > 0$ and $E_n \in \text M$, then $A \in \text M$ and $\mu (A) > 0$.
For a), I tried to proof that if $\mu^* (A) > 0$, we can get all $A_n=A \cap E_n$ with null outer measure out without affecting the outer measure of $A$ and we can prove that then $\mu^*(X)= \infty$, but I'm not sure if that comes anywhere. For b), I'm completely stuck.
If someone could lend me a hand about it, I would be very grateful.
Hints: For (a), you want to look at re-writing A in terms of the lim sup of a set and then applying subadditivity.
For (b), try making the sets disjoint and then using additivity (it follows from the caratheodory condition, you might want to prove that).
I can be more specific if you're still stuck, but this should be a good direction to start in.