# On Lebesgue outer measure

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)

• 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. – Faraad Armwood Sep 17 '17 at 12:54
• 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.) – GEdgar Sep 17 '17 at 13:07