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Let $X$ be any set. Let $\mathcal{C}$ be the collection of singleton points $\{x\}$ , $x\in X$. Let $m$ be the set function defined on $\mathcal{C}$ with $m(\{x\})=1.$ I wish to be able to find the following and I need help/guidance:

(a) find the outer measure $m^*$ induced by $m$.
(b) find the sigma algebra of measurable sets.
(c) find the measurable functions if $\bar{m}$ is the Caratheodory measure induced by $m$.
(d) determine whether $(X,A,\bar{m})$ is a complete measure space, where $A$ is the sigma algebra of measurable sets.

Thanks.

For (a) I get that $m^*(\emptyset)=0$. If $E\subset \mathcal{C}$, $E$ finite, then $m^*(E) =$ number of elements in $E$. If $E$ is infinite, then $m^*(E) = \infty$. Is this right?

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    Congratulations, you did (a) yourself.2012-02-28

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For (b), recall that $A\subseteq X$ is measurable if $\mu^*(E)=\mu^*(E\cap A)+\mu^*(E\cap A^c)$ for all $E\subseteq X$. Now, we proceed by cases: if $E$ is finite then $\mu^*(E)=|E|=|E\cap A|+|E\cap A^c|=\mu^*(E\cap A)+\mu^*(E\cap A^c)$. On the other hand, if $E$ is infinite then either $E\cap A$ or $E\cap A^c$ are infinite, in either case both sides of the equation are infinite so the equality holds. Hence the $\sigma$-algebra of measurable sets is equal to the power set of $X$. The rest follows easily from this.

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    @GEdgar: Since the power set is the sigma algebra of measurable sets, any function $f:X\to \bar{R}$ is measurable since \{x:f(x) for any real number $a$, is in the power set.2012-02-28