By W. Thirring

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Therefore, we have only to choose those corresponding to m odd. For any such m fixed the open sets O(m−1)/2n and O(m+1)/2n have been selected. Since {X; U} is normal, there exists an open set Om/2n such that O¯ m−1 ⊂ O 2mn ⊂ O¯ 2mn ⊂ O m+1 . 2n 2n Define f : X → [0, 1] by setting f (x) = 1 for x ∈ B and f (x) = inf{t x ∈ Ot } for x ∈ X − B. By construction f (x) = 0 on A and f (x) = 1 on B. It remains to prove that f is continuous. From the definition of f it follows that for all s ∈ (0, 1] [f < s] = t~~s Ot . ~~

579 Index . . . . . . . . . . . . . . . . . . . . . . . . 585 16 17 574 576 Chapter 1 Preliminaries 1 Countable Sets A set E is countable if it can be put in one-to-one correspondence with a subset of the natural numbers N. Every subset of a countable set is countable. 1 The set S E of the finite sequences of elements of a countable set E is countable. Proof Let {2, 3, 5, 7, 11, . . , m j . } be the sequence of prime numbers. Every positive integer n has a unique factorization, of the type α n = 2α1 3α2 · · · m j j where the sequence {α1 , .

1. . . . . . The Sharp Maximal Function . . . . . . . . . . Proof of the Fefferman–Stein Theorem . . . . . . . The Marcinkiewicz Interpolation Theorem. . . . . . 1 Quasi-linear Maps and Interpolation . . . . . Proof of the Marcinkiewicz Theorem . . . . . . . . Rearranging the Values of a Function . . . . . . . . Some Integral Inequalities for Rearrangements . . . . . 1 Contracting Properties of Symmetric Rearrangements . . . . . . . . .