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Let $X$ be a set and $\mathscr{A}$ a collection of subsets of $X$ that form an algebra of sets. Suppose $l$ is a measure on $\mathscr{A}$ such that $l(X) < \infty$. Define $\mu^{*} $ as $$ \mu^{*}(E) = \inf \; \{\Sigma \, l(C_{i}) \;\; | \;\; C_{i} \in A \; , E \subset \cup_{i=1}^{\infty}C_{i} \}. $$ Prove that a set $A$ is $\mu^{*}$ measurable if and only if $$ \mu^\star(A) = l(X) - \mu^\star(A^{c}) $$

Can I use the fact that by definition, if $A$ is $\mu^{*}$-measurable, then for all $B\subset X$,

$$ \mu*(B)=\mu*(B\cap A)+\mu*(B\cap A^c) $$

What is the relationship between $\mu^{*}(X)$ and $l(X)$, I'm very unsure about this chapter (constructing measure).

Thanks for any input

AJY
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Akaichan
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1 Answers1

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Don't overthink it. Yes, you can use the Caratheodory criterion $\mu^*(B) = \mu^*(B\cap A) + \mu^*(B\cap A^c)$.

Try writing $\mu^*(A) + \mu^*(A^c) = l(X)$ instead. Does it give you any ideas?

Gyu Eun Lee
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  • I actually I did do that, that's why I asked about the relationship of $\mu*(B)$ and $l(X)$ :( – Akaichan Nov 20 '14 at 19:55
  • You seem to have made a mistake in the definition of $\mu^*(E)$. Are you sure there isn't a $l(C_i)$ somewhere? – Gyu Eun Lee Nov 20 '14 at 20:09
  • Right, I mistyped it – Akaichan Nov 20 '14 at 20:30
  • I mean what I have problem on is the reverse direction, the forward is almost instant from the definition, for converse, I have to show that if $l(X)=\mu(A)+\mu(A^c)$, then for all $B\subset X$, $\mu(B)=\mu(B\cap A)+ \mu*(B\cap A^c)$ – Akaichan Nov 21 '14 at 00:52