Proposition 12 :-E be any finite outer measure. then is disjoint collection of open intervals

Here we see the proof of theorem 12 on HL ROYDEN
Let E be  a finite measurable set of finite outer measure. Then for each epsilon greater than zero there is a finite disjoint collection of open intervals  Ik for which O= union of Ik then
m*(E-O)+m*(O-E) less than epsilon.
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