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For a set A of real numbers $(A \subset \mathbb{R}) $, define outer measure of set A as, $ m^*(A) = \inf \{\sum_{k=1}^{\infty}l (I_k)/A\subseteq \cup_{k=1}^{\infty}I_k\}. $ I want to find the outer measure of set $(2,5) $ by using of definition of outer measure. How to apply definition of outer measure? (I know outer measure of an interval is its length but i want to know how to use definition of outer measure to calculate outer measure of given set)

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  • $\begingroup$ If you read the proof that outer measure of an interval is just its length, there they will explicitly show how to pick the cover to give you the result. $\endgroup$ – Faraad Armwood Sep 17 '17 at 12:54
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    $\begingroup$ What is needed is the interesting but nontrivial fact: if you cover $(2,5)$ by any sequence of intervals, then the sum of the lengths is at least $7$. (To do this, I would use the Heine-Borel theorem.) $\endgroup$ – GEdgar Sep 17 '17 at 13:07

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