I'm trying to read this proof, and I'm stuck on the inequality on page 27 following the statement "It follows that every measurable subset..." Why does it hold?(adsbygoogle = window.adsbygoogle || []).push({});

The theorem is about signed measures, i.e. functions that are like measures, but can assign both positive and negative "sizes" to sets. A measurable set is said to bepositiveif all its measurable subsets have a non-negative size. The termnegativeis defined similarly. The theorem asserts that the set X is a disjoint union of a positive set A and a negative set B. The strategy of the proof is roughly this: First find a set B and show that it's negative. Define A=X-B. Suppose that A is not positive. (This will lead to a contradiction). It's not too hard to see that A doesn't have any negative subsets, but we can still pick a subset [itex]E_0\subset A[/itex] that has a negative size. Then we cut away disjoint pieces of [itex]E_0[/itex], denoted by [itex]E_1,E_2,\dots[/itex] that have positive sizes. The goal is to show that [itex]E_0-\bigcup_{k=1}^\infty E_k[/itex] is negative.

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# Proof of Hahn decomposition theorem

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