Define the the outer measure of a subset of the unit interval as the infimum of the sum of lengths of intervals over all such coverings by intervals. The inner measure of the same subset is one less the outer measure of the complement of the given subset. Then we have a theorem: the inner measure is less than or equal to the outer measure for any subset of the unit interval.
The proof in my book argues by contradiction. It obtains, by definition of outer measure, a covering (by intervals) of the given subset and another covering of its complement such that the sum of the sum of the lengths of the intervals (in each cover) adds less than 1. It then takes the union of such coverings and obtains the contradiction that the unit interval is contained in the new covering, yet the measure of the sums of lengths of the intervals in this new covering adds to less than 1 by the previous statement, a contradiction to a previous theorem (that for elementary sets, or intervals, if a set is contained in a union then the measure of the subset is less or equal to the sum of measures in the interval)
My problem is the theorem contradicted was applicable for only elementary sets i.e. finite disjoint unions of intervals (representation by such a union may not be unique). The union of the coverings obtained at the end of the proof is by no means an elementary set, since the coverings obtained by the def of outer measure are not necessarily disjoint. In fact my book clearly uses the measure defined on intervals for this union, and there's no reason that the union of two intervals (from the two coverings) MUST be an interval.
I understand given a finite union of intervals I can rewrite this as a disjoint union of intervals (by suitably arranging the endpoints, discarding repetitions, taking open intervals between the endpoints and the degenerate singleton sets, where we allowed such to be considered intervals). Then measure is defined for this new union. How is it that the measure of this new disjoint representation must coincide with the "measure" on the original union? The original could be bigger.
Thanks in advance for any clarification or comments, and i apologize for describing everything in plain English, if anyone is averse to that.