# hw4 - Math 245C Homework 4 Brett Hemenway June 1 2007...

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Unformatted text preview: Math 245C Homework 4 Brett Hemenway June 1, 2007 Folland Chapter 7 4. Let X be an LCH space. (a) Suppose f ∈ C c ( X, [0 , ∞ )). Then f- 1 ([( a, ∞ )) is closed since f is continuous, and f- 1 ([ a, ∞ )) is compact since it is a closed set contained in the support of f which is compact. Now, f- 1 ([ a, ∞ )) = f- 1 ∞ n =1 ( a- 1 n , ∞ ) ! = ∞ n =1 f- 1 (( a- 1 n , ∞ )) . But f- 1 (( a- 1 n , ∞ )) is open for each n since f is continuous, thus f- 1 ([ a, ∞ )) is the countable intersection of open sets. (b) Let K be a compact G δ set, i.e. K = T ∞ n =1 U n , where the U n are open. Then by the Locally Compact version of Urysohn’s Lemma (Lemma 4.32), We can find an f n such that 0 ≤ f n ≤ 1, and f n | K = 1, and f n = 0 outside a compact subset of U n . Let f = ∞ X n =1 1 2 n f n . Then 0 ≤ f ≤ 1 and if x ∈ K , f ( x ) = ∞ X n =1 1 2 n f n ( x ) = ∞ X n =1 1 2 n = 1 . On the other hand, if x 6∈ K , then there exists an n such that x 6∈ U n , thus f n ( x ) = 0, which means f ( x ) ≤ 1- 1 2 n < 1. So f- 1 ( { 1 } ) = K . 1 11. Suppose μ is a Radon measure on X such that μ ( { x } ) = 0 for all x ∈ X , and A ∈ B X is such that 0 < μ ( A ) < ∞ . Since μ is Radon, for any α < μ ( A ), we can find a compact set K ⊂ A such that α < μ ( K ), by possibly replacing A with K , we may assume that A is compact. For each x ∈ A , we have μ ( { x } ) = 0. Since μ is outer regular, for any > 0, we can find an open neighborhood E x of x such that μ ( E x ) < . Now, S x ∈ A E x covers A , and since A is compact, it has a finite subcover E x 1 ,...,E x n . Now, for any α < μ ( A ), we can find an n such that α- ≤ μ n [ i =1 E x i ! ≤ α To find a set with measure exactly α , we repeat this construction. Using the preceding construction, find E 1 such that α- ≤ μ ( E 1 ) ≤ α . Now, find E 2 ⊂ E c 1 such that 2 ≤ μ ( E 2 ) ≤ α- μ ( E 1 ), continuing, we can find E n ⊂ ( E 1 ∪ ··· ∪ E n- 1 ) c such that 2 n- 1 ≤ μ ( E n ) ≤ α- μ n- 1 [ i =1 E i !...
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hw4 - Math 245C Homework 4 Brett Hemenway June 1 2007...

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