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I am reading lecture notes I borrowed from friend. It defines the filter of subsets of $[0,1]$ with Lebesgue measure $1$. Then it defines sets $A$ to be called $F$-stationary iff $A \cap Y \neq \varnothing$ for all $Y$ in the filter. Then it writes the $F$-stationary sets are the sets of positive outer measure.

Why is it outer measure and not Lebesgue measure? And why is it true? Does it hold that all subsets of $[0,1]$ with cardinality equal the cardinality of $[0,1]$ must have positive (outer or Lebesgue) measure?

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up vote 6 down vote accepted

First the easy part, the Cantor set is of the same cardinality as $[0,1]$ and has Lebesgue measure zero.

Now, we want to define a notion of stationarity on all subsets, not just measurable sets. Sets which are not Lebesgue measurable cannot have outer measure zero (because the Lebesgue measure is complete).

Finally, suppose $E$ is any subset, and $A$ is measurable with measure $1$, such that $m^*(E\cap A)=0$ then $m^*(E)=m^*(E\setminus A)\leq m^*([0,1]\setminus A)=0$.

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Beat me to it... –  Arthur Fischer Jan 28 '13 at 12:10
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I can't find a funny comeback. I'm going to cook lunch, and I'll get back to you with a funny remark if I come up with one. –  Asaf Karagila Jan 28 '13 at 12:12
    
Why can a set that is not Lebesgue measurable not have outer measure zero? –  dolan Jan 28 '13 at 13:17
    
@user58975: Having outer measure zero means that you're a subset of a measure zero Borel set. But the Lebesgue measure is defined as adding all the subsets of measure zero Borel sets. It follows that outer measure zero implies measurability, and therefore measure zero. –  Asaf Karagila Jan 28 '13 at 13:20
    
I am sorry: why does outer measure zero imply is subset of measure zero Borel set? –  dolan Jan 28 '13 at 13:21
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