This problem is about the construction of Borel measure on the real line.(in Folland chapter 1,section 5.), more specifically, construction of premeasure on h interval.
To prove that
$\mu_{0}\left(\bigcup_{1}^{n}\left(a_{j},b_{j}\right]\right)=\sum_{1}^{n}\left[F\left(b_{j}\right)-F\left(a_{j}\right)\right]$ is a premeasure for finite disjoint h interval .
The idea is first approximate those h interval ($a_i$,$b_i$] that form (a,b] by $[a_i +\delta,b_i]$ and since compact set has finite open cover,then we can use "size" of finite open cover to approixmate the "size" of (a,b].
My question is why finite open cover is necessary in the proof,why can't we just use infinite open cover $(a_i,b_i+\delta)$ for all component of (a,b] to approixmate "size" of (a,b],More clearly :
$\mu_0(I) \le F(b)- F(a+\delta) +\epsilon \le F(b+\delta)-F(a_1) +\epsilon \le \sum_{i\to \infty}(F(b_i+\delta)-F(a_i)) +\epsilon \le \sum_{i\to \infty} \mu_0(I_j) + 2 \epsilon$?
The third ineuqality is due to the incresing of function F.
Do I state the problem clear?