i)Describe the outer measure $u^*$ on $2^{\Omega}$ induced by the given $\mu$ ii)describe M($u^{*}$) and determine if M($u^*$) is a $\sigma$-field, and check to see whether $u^*$ = u on the given collection A.
1)Let $\Omega$ be countable infinite and let B denote the field of sets such that A is finite or $B^c$ is finite, and write $\mu(A) = 0$ if A is finite and $\mu(A) = 1$ if $A^c$ is finite.
2)Redo (1) assuming that $\Omega$ is uncountable.
My try I can see that there are two cases too consider either A is finite or A is infinite . It seems to me that according to my intuition that $u^*(A) = 0$ if A is finite or $u^*(A) = 1$ if A is infinite, however I can't seem to think of argument to justify this claim.
If someone could just give me some hint for this question that would be great.
(1) The set function $\mu$ is not defined on the sets $C$ for which both $C$ and $C^c$ are infinite. As far as the outer measures: if $C\notin $ B then $\mu^*(C)=\mu^*(C^c)=1$ and if $A\in \text{B}$ then $\mu^*(A)=\mu(A)$. The purpose of the extension is to define the measure of some sets outside B. A $C\notin$ B is (Carathéodory) measurable if for all $A\in\text{B}$ $$\mu^*(A)=\mu^*(A\cap C)+\mu^*(A\cap C^c).$$ Fix one $e\in \Omega$ and consider a special $A_e=\Omega \setminus \{e\}\in \text{B}$. $A_e $ will have infinitely many elements common with $C$ and common with $C^c$ for all $C$ of the kind we are interested in. For any such $C$ then $$1=\mu^*(A_e)\not=\mu^*(A_e\cap C)+\mu^*(A_e\cap C^c)\ge2.$$That is, none of the sets outside B can be measurable.
Remarque
I could have said right at the beginning that $\mu$ was not $\sigma$-additive. Let $\Omega=\mathbb N $, $A_2=\{2\}$, $A_3=\{3\}$, and so on. All these sets are finite, so $\mu(A_i)=0$ for all $i\gt1$. These sets and their complements are in B. Furthermore, their union: $A=\{2,3,4,...\}\in \text{B}$ because $A^c=\{1\}$. But $1=\mu(A)\not=0+0+...=0$.
(2) The argumentation above can be copied here.