chap10 - further integration

chap10 - further integration - Further investigation of...

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Unformatted text preview: Further investigation of integration 1 Chapter 10 Further investigation of integration We have two exceedingly important things to discuss in this chapter: integration over general subsets of R n , and change of variables. We first discuss a characterization of contented sets. A. Topological background We have often hinted at many of the following concepts. It is now time to make certain we understand them completely, and have useful names and notations for them. Recall the definition from Section 3A: if A is a subset of R n and x ∈ A , we say that x is an interior point of A if there exists r > 0 such that the ball B ( x ,r ) ⊂ A . A B(x,r) DEFINITION. The set of all interior points of A is called the interior of A , and is denoted int A or int( A ). As we have defined a set to be open if and only if all its points are interior points, we see that A is open ⇐⇒ A = int A . Recall that we proved in Section 3A that the open ball B ( x,r ) is itself an open set. PROBLEM 10–1. Prove that every point of int A is in the interior of int A . That is, int(int A ) = int A . PROBLEM 10–2. The preceding problem shows that int A is an open set. Prove that it is the largest open subset of A , in the sense that if B is an open set and B ⊂ A , then B ⊂ int A . 2 Chapter 10 PROBLEM 10–3. Prove that int( A ∪ B ) ⊃ int A ∪ int B ; int( A ∩ B ) = int A ∩ int B. Prove that the inclusion expressed above may be a strict one. PROBLEM 10–4. Prove that the union and the intersection of two open sets are also open sets. The notion which is dual to interior will now be discussed. DEFINITION. Let A ⊂ R n and x ∈ R n . Then x is a closure point of A if for every < r < ∞ , B ( x ,r ) ∩ A is not empty. A B(x,r) DEFINITION. The set of closure points of A is called the closure of A , and is denoted cl A or cl( A ). DEFINITION. A set is said to be closed if it contains all its closure points. PROBLEM 10–5. Prove that cl(cl A ) = cl A . PROBLEM 10–6. Prove that the closed ball B ( x,r ) is a closed set. PROBLEM 10–7. Devise the analog of Problem 10–2 and prove the result. Further investigation of integration 3 PROBLEM 10–8. Devise the analog of Problem 10–3 and prove the results. PROBLEM 10–9. Prove that the union and the intersection of two closed sets are also closed sets. PROBLEM 10–10. Interior and closure are truly dual notions, for int( R n- A ) = R n- cl A. PROBLEM 10–11. Prove that A is open ⇐⇒ R n- A is closed. WARNING . Don’t misread this problem. A frequent error is to think that a set is open ⇐⇒ it is not closed. We have discussed int A and cl A . A third set will be of great interest to us: DEFINITION. For a given set A ⊂ R n , the boundary of A is the set bd A = bd( A ) = cl A- int A....
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This note was uploaded on 09/28/2008 for the course MATH 1220 taught by Professor Hatcher during the Fall '08 term at Cornell.

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chap10 - further integration - Further investigation of...

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