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Unformatted text preview: 1 Measure 1.1 Rectangles Almost Disjoint. A union of rectangles is said to be almost disjoint if the interiors of the rectangles are disjoint. Lemma 1.1. If a rectangle is the almost disjoint union of finitely many other rectangles, say R = S N k =1 R k , then  R  = ∑ N k =1  R k  . Lemma 1.2. If R,R 1 ,...,R N are rectangles and R ⊂ S N k =1 R k , then  R  ≤ ∑ N k =1  R k  . Theorem 1.3. Every open subset O of R can be written uniquely as a countable union of disjoint open intervals. Theorem 1.4. Every open subset O of R d , d ≥ 1, can be written as a countable union of almost disjoint closed cubes. 1.2 Outer Measure Exterior Measure. For any subset E ∈ R d , the exterior measure of E is m * ( E ) = inf ∑ ∞ j =1  Q j  , where the infimum is taken over all countable coverings E ⊂ S ∞ i =1 Q j . Observation 1 (Monotonicity). If E 1 ⊂ E 2 , then m * ( E 1 ) ≤ m * ( E 2 ). Observation 2 (Countable subadditivity). If E = S ∞ i =1 E j , then m * ( E ) ≤ ∑ ∞ i =1 m * E j . Observation 3. If E ⊂ R d , then m * ( E ) = inf m * ( O ), where the infimum is taken over all open sets O containing E . Observation 4. If E = E 1 ∪ E 2 , and d ( E 1 ,E 2 ) > 0, then m * ( E ) = m * ( E 1 ) + m * ( E 2 ). Observation 5. If a set E is the countable union of almost disjoint cubes E = S ∞ i =1 Q j , then m * ( E ) = ∞ X i =1  Q j  . 1.3 Lebesgue Measure Measurable Set. A subset E is Lebesgue measurable if for any > 0 there exists an open set O with E ⊂ O and m * ( O  E ) < . If E is measurable, we define its Lebesgue measure to be m ( E ) = m * ( E ). Property 1. Every open set in R d is measurable. Property 2. If m * ( E ) = 0, then E is measurable. In particular, if F is a subset of a set of exterior measure 0, then F is measurable. Property 3. A countable union of measurable sets is measurable. Property 4. Closed sets are measurable. Property 5. The complement of a measurable set is measurable. Property 6. A countable intersection of measurable sets is measurable. Lemma 3.1. If F is closed, K is compact, and these sets are disjoint, then d ( F,K ) > 0. Theorem 3.2. If E 1 ,E 2 ,... , are disjoint measurable sets, and E = S ∞ i =1 E j , then m ( E ) = ∑ ∞ i =1 m ( E i ). Corollary 3.3. Suppose E 1 ,E 2 ,... are measurable subsets of R d . 1. If E k % E , then m ( E ) = lim N →∞ m ( E N ). 2. If E k & E and m ( E k ) < ∞ for some k , then m ( E ) = lim N →∞ m ( E n ). Theorem 3.4. Suppose E is a measurable subset of R d . Then, for every > 0: 1. There exists an open set O with E ⊂ O and m ( O  E ) ≤ . 2. There exists a closed set F with F ⊂ E and m ( E F ) ≤ . 3. If m ( E ) is finite, there exists a compact set K with K ⊂ E and m ( E K ) ≤ ....
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This note was uploaded on 04/01/2008 for the course MATH 501 taught by Professor Wysocki,krzysztof during the Fall '08 term at Penn State.
 Fall '08
 WYSOCKI,KRZYSZTOF
 Math, Angles

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