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This is Exercise 4.15 from "Real Analysis for Graduate Students": Let $X$ be a set and $A$ a collection of subsets of $X$ that form an algebra of sets. Suppose $l$ is a pre-measure on $A$ such that $l(X)<\infty$. Define $\mu^*(E):=inf \{\sum_{i=1}^{\infty}l(A_i): each\space A_i\in A, E\subset\cup_{i=1}^{\infty}A_i\}$ for any subset $E$ of $X$. Prove that a set $B$ is $\mu^*$-measurable iff $\mu^*(B)=l(X)-\mu^*(B^c)$.

I first want to remind the definition that a set $B$ is $\mu^*$-measurable iff $\mu^*(E)=\mu^*(E\cap B)+\mu^*(E\cap B^c)$ for any arbitrary subset $E$ of $X$. It's easy to show that $l(X)=\mu^*(X)$, so the expression given in the problem becomes $\mu^*(B)+\mu^*(B^c)=\mu^*(X)$. One direction is obvious: If you take the set $E$ in the definition of $\mu^*-$measurable as $X$, $\mu^*(B)+\mu^*(B^c)=\mu^*(X)$ directly comes. However, I couldn't make much progress for the other direction. Thanks!

vgmath
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    Here you would call $l$ a pre-measure. – Math1000 May 28 '15 at 01:11
  • Yes, actually you're right, I think standard terminology is pre-measure. But, since the book I mentioned above defines it in the way I wrote in the P.S part, I somehow didn't change it, and took as it is. Sorry about that, I'll edit right away. – vgmath May 28 '15 at 01:50
  • The reverse direction seems to be a little tricky! - a sketch is given here – Lost in a Maze May 28 '15 at 05:29

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