MATH501_Study_Guide - 1 1.1 Measure Rectangles Almost...

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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 can be written uniquely as a countable union of disjoint open intervals. Theorem 1.4. Every open subset O of 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 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 sub-additivity). If E = S i =1 E j , then m * ( E ) i =1 m * E j . Observation 3. If E 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 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 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 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 ) . 4. If m ( E ) is finite, there exists a finite union F = S N j =1 Q j of closed cubes such that m ( E 4 F ) . Corollary 3.5. A subset E of d is measurable 1. if and only if E = V \ N where V is a G δ -set and m ( N ) = 0, 2. if and only if E = H M where H is an F σ -set and m ( M ) = 0. 1
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1.4 Measurable Functions Characteristic Function. The characteristic function of a set E is χ E ( x ) = 1 , x E 0 , x / E .
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