Posted by A on November 13, 2007, 9:42 pm, in reply to "Inner measure?"
Usually one defines inner measure as follows: m^*(O):=sup{m(O_i), O_i measurable and O_i \subset O}
Inner measure would not have the nice properties of the outer measure i.e. countable sub-additivity need not hold.
Your definition of inner measure seems to be "corrupt". You want the O_i's to be disjoint, right? Else you'd get \infty for many of your sets. Also, if the O_i's are not cubes then how would you define |O_i|?
If you want to define the "inner measure" using disjoint cubes O_i you'd still get a different "measure":
Consider the fat cantor set C, defined in SS Ch.1, p. 40, Exercise #4. We proved that m(C)>0 and that C does not contain any open intervals. It follows that m^*(C)=0.
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