Reading Lebesgue outer measure of Lebesgue Measure Chapter from Carothers' Real Analysis and some properties and their proofs are here:
Basically, I can't understand the proof for reverse inequality of (vi). I've no ideal how to explain the "why" there. Besides, what's the relationship between $[a_n,b_n]$ and $I_n$ or $J_n$(Is $I_n$ picked up from the set of {$[a_n,b_n]$ | $n∈ \mathbb N$} )?
Thanks all.
Add:
Definition of Carothers' Outer Measure
The reverse inequality of (vi) is: $$m^*(E)\ge{\rm inf}\left\{\sum_{n=1}^\infty(b_n-a_n) : \, E\subset\bigcup_{n=1}^\infty(a_n,b_n)\right\} .$$ The reason for (why?) is: if the LHS is infinity, there is nothing to prove.
As for the rest of the proof of (vi), the collection of intervals $(I_n)$ exists by definition of outer measure. (I'm assuming that the Carothers definition of outer measure doesn't require the intervals $(I_n)$ to be open--they can be closed, half-open, etc.) Then the $(J_n)$ are selected to be open intervals $(a_n,b_n)$ such that each $I_n\subset J_n$ and $J_n$ is slightly longer than $I_n$. It is always possible to find such ($J_n$). For example, the closed interval $[u,v]$ is contained within the open interval $(u-\delta,v+\delta)$.
Note: (vi) is saying that in the definition of outer measure you can require each covering interval $I_n$ to be open. In some treatments of measure theory, outer measure is defined with that requirement right off the bat.
EDIT: Just saw the edit to your OP and it confirms my assumption of the Carothers def of outer measure.