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- TL;DR Summary
- I'm stuck at a simple idea in Proposition 1.15 in Folland's real analysis text, where he makes an assumption that I'm trying to grasp rigorously.

Below is Proposition 1.15 in Folland at the beginning of the section of Borel measures on ##\mathbb R## (he is trying to construct a measure from ##F##). Here the algebra ##\mathcal{A}## is the finite disjoint union of h-intervals, where h-interval is a set of the form ##(a,b]##, ##(a,\infty)## or ##\varnothing## for ##-\infty\leq a<b<\infty##.

I assume the reader knows what a premeasure is (more or less a set function on an algebra that maps ##\varnothing## to ##0## and is countably additive for any sequence of disjoint sets in the algebra). First Folland shows that ##\mu_0## is well-defined. Then he goes on to show countable additivity, and in doing so, he makes a statement that I understand intuitively but that I don't understand how to write down rigorously.

Given a sequence of disjoint h-intervals ##\{I_j\}_1^\infty## such that ##\bigcup_1^\infty I_j\in\mathcal{A}##, he makes the following claim.

Again, intuitively this makes sense, but I'm looking for an explanation with symbols (e.g. where one proves that the union of the intervals in each subsequence is in fact a single h-interval). How would you do this?

I assume the reader knows what a premeasure is (more or less a set function on an algebra that maps ##\varnothing## to ##0## and is countably additive for any sequence of disjoint sets in the algebra). First Folland shows that ##\mu_0## is well-defined. Then he goes on to show countable additivity, and in doing so, he makes a statement that I understand intuitively but that I don't understand how to write down rigorously.

Given a sequence of disjoint h-intervals ##\{I_j\}_1^\infty## such that ##\bigcup_1^\infty I_j\in\mathcal{A}##, he makes the following claim.

Since ##\bigcup_1^\infty I_j## is a finite disjoint union of h-intervals, the sequence ##\{I_j\}_1^\infty## can be partitioned into finitely many subsequences such that the union of the intervals in each subsequence is a single h-interval.

Again, intuitively this makes sense, but I'm looking for an explanation with symbols (e.g. where one proves that the union of the intervals in each subsequence is in fact a single h-interval). How would you do this?