Here is the definition of Carothers' Lebesgue Outer Measure:
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And countably subadditive property of Lebesgue Outer Measure has been talked here:
I can understand all proofs. However, I'm wondering why the set $G$ in Corollary 16.7 is necessarily open there? How about a closed set $G$? Similarly, see the proof following Corollary 16.7. Why should we necessarily find a sequence of open intervals ($I_n$)? How about a closed one or a neither open nor closed set?
Appreciate much.