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JHiN8y-12

# JHiN8y-12 - 30 SPRING 2012 12 Lebesgue Measure Open Sets...

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30 SPRING 2012 12. Lebesgue Measure: Open Sets and Compact Sets So far we have defined λ ( A ) for A empty, for any closed box A , and more generally for any special polygon A R n . Definition 12.1. If G R n is a nonempty open set, let λ ( G ) = sup { λ ( P ) | P is a special polygon such that P G } . Since every nonempty open set contains an open ball in R n , and every open ball contains a nondegenerate closed box, we have 0 < λ ( G ) for any nonempty open set G . Examples 12.2. (i) λ ( R n ) = . To see this, fix M > 0 and consider the closed box I M = [0 , M ] × [0 , 1] × · · · × [0 , 1]. Then I M is a special polygon and I M R n , λ ( R n ) λ ( I M ) = M . Since M > 0 was arbitrary, we have λ ( R n ) = . (ii) If G is a bounded nonempty set, then λ ( G ) < . To see this, by definition there exists K R such that G B K ( 0 ). So if we let J K be the closed box [ K, K ] n , then G J K . For any special polygon P with P G we thus have P J K , and therefore λ ( P ) λ ( J K ) = (2 K ) n < . Taking the sup over all such P ’s gives λ ( G ) (2 K ) n < . Exercise 12.1. Show that for any special polygon P , we have λ ( P ) = λ ( P o ), where P o denotes the interior of P (that is, the largest open set contained in P ). Hint: First consider a closed box I , and argue that for any > 0, we can construct a closed box J such that J I 0 I and λ ( I ) λ ( J ) < . Proposition 12.3. Let G i R n be open for each i N . (i) If G 1 G 2 , then λ ( G 1 ) λ ( G 2 ) . (ii) λ ( i =1 G i ) i =1 λ ( G i ) , with equality holding if the G i are ( pairwise ) disjoint.

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JHiN8y-12 - 30 SPRING 2012 12 Lebesgue Measure Open Sets...

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