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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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  • $\begingroup$ It is customary to show your attempts, rather than asking us to do it for you. Start with (a). Do you know the definition of $m^*$? If so, what do you get in this case? $\endgroup$
    – GEdgar
    Commented Feb 28, 2012 at 4:09
  • $\begingroup$ @GEdgar: I've added to my post. Is what Ive done right? $\endgroup$
    – Cindy
    Commented Feb 28, 2012 at 4:34
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    $\begingroup$ Congratulations, you did (a) yourself. $\endgroup$
    – GEdgar
    Commented Feb 28, 2012 at 15:33

1 Answer 1

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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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  • $\begingroup$ Is what I did for (a) right? $\endgroup$
    – Cindy
    Commented Feb 28, 2012 at 6:36
  • $\begingroup$ Also, will this be right for (c); $\bar{m}:2^X\to [0,\infty]$. for (d), can I say $(X,A,\bar{m})$ is complete since it measure space is induced by an outer measure? $\endgroup$
    – Cindy
    Commented Feb 28, 2012 at 7:05
  • $\begingroup$ You can say that, yes. But in this case (since you know the measure and its sigma-algebra explicitly) why not check the definition of "complete"? $\endgroup$
    – GEdgar
    Commented Feb 28, 2012 at 15:33
  • $\begingroup$ @GEdgar: Thanks for your comment. Yes I can show completeness. However, I'm a little unsure about what the measurable functions are. $\endgroup$
    – Cindy
    Commented Feb 28, 2012 at 17:25
  • $\begingroup$ The sigma-algebra of measurable sets is the power set. So what do you think the measurable functions are? What is the definition of "measurable function"? $\endgroup$
    – GEdgar
    Commented Feb 28, 2012 at 17:54

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