Prove that if $F$ is a finite collection of open intervals that covers a compact interval $[a, b]$, then the sum of the lengths of the intervals in the collection is strictly greater than $b − a$
This can be proved without considering measure theory. But I am looking for a proof in terms of Lebesgue measure.
Please suggest.
It suffices to show that: Let $U$ be an open set which contains $[a, b]$, then $m(U) > b-a$. (Let $U$ be the union of your open intervals in $F$, then the quantity you want is $\ge m(U)$.
As $[a, b] \subset U$, we have $m(U) \ge b-a$. To show that we have strict equality, it suffices to show that $m(U\setminus [a, b])>0$. But as $U\setminus [a, b]$ is open and nonempty (as $[a, b]$ is not open). It suffices to show that $m(W) >0$ for all open nonempty sets. But that is quite obvious.