Real and Complex Analysis (Rudin)

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POSITIVE BOREL MEASURES 39 will mean that K is a compact subset of X, thatf e Cc(X), that 0 5;f(x) 5; 1 for all x e X, and thatf(x) = 1 for all x e K. The notation f-< V (2) will mean that V is open, thatf E Cc(X), 0 5;f 5; 1, and that the support of! lies in V. The notation will be used to indicate that both (1) and (2) hold. K -
40 REAL AND COMPLEX ANALYSIS f(x) = 1 if x E K, and that f has its support in Vo. The proof will be completed by showing thatf = g. The inequality f..(x) > g.(x) is possible only if r > s, x E v,., and x ¢ v.. But r > s implies v,. c V.. Hence f.. ::; g. for all rand s, so f::; g. Suppose f(x) < g(x) for some x. Then there are rationals rand s such that f(x) < r < s < g(x). Since f(x) < r, we have x ¢ v,.; since g(x) > s, we have x E v.. By (3), this is a contradiction. Hencef = g. IIII 2.13 Theorem Suppose V1, ... , v" are open subsets of a locally compact Hausdorff space X, K is compact, and K c V 1 U ... U V n • Then there exist functions hi -< V; (i = 1, ... , n) such that (x E K). (1) Because of (1), the collection {h1' ... , hn} is called a partition of unity on K, subordinate to the cover {V1' ... , v,,}. PROOF By Theorem 2.7, each x E K has a neighborhood w" with compact closure w" c V; for some i (depending on x). There are points Xl' ••• , Xm such that w,,1 U ... U w"m :::;) K. If 1::; i ::; n, let Hi be the union of those w"j which lie in V;. By Urysohn's lemma, there are functions gi such that Hi -< gi -< V;. Define h1 = gl h2 = (1 - gdg2 h n = (1 - gl)(1 - g2) ... (1 - gn-1)gn' Then hi -< V;. It is easily verified, by induction, that (2) Since K c H1 U ••. U H n , at least one g~x) = 1 at each point x E K; hence (3) shows that (1) holds. IIII The Riesz Representation Theorem 2.14 Theorem Let X be a locally compact Hausdorff space, and let A be a positive linear functional on CiX). Then there exists a a-algebra 9Jl in X which contains all Borel sets in X, and there exists a unique positive measure Jl. on 9Jl which represents A in the sense that (a) Af = Ix f dJl.for every f E Cc(X), and which has the following additional properties:
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REAL AND COMPLEX ANALYSIS
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BIBLIOGRAPHY 1. L. V. Ahlfors: "Com
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LIST OF SPECIAL SYMBOLS AND ABBREVI
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INDEX Absolute continuity, 120 of f
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INDEX 411 Double integral, 165 Dual
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INDEX 413 Lower limit, 14 Lower sem
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INDEX 415 Separable space, 92, 247
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