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7fFH0H-07 - ASSIGNMENT 7 SOLUTIONS MAT 473 SPRING 2012...

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ASSIGNMENT 7 · SOLUTIONS MAT 473 · SPRING 2012 Exercise 12.2. Prove that if x R n and P is a special polygon in R n , then λ ( P ) = λ ( P + x ). Conclude that λ ( G ) = λ ( G + x ) for every open set G in R n . Solution. It is clear that I + x is a closed box if and only if I is, in which case λ ( I + x ) = λ ( I ). Now suppose P is a special polygon, and (by Lemma 11.6) write P = k i =1 I i as a finite union of nonoverlapping nondegenerate closed boxes. Then it is easy to see that P + x = k i =1 ( I i + x ) is a finite union of nonoverlapping nondegenerate closed boxes. Hence P + x is also a special polygon, and by the above (and Lemma 11.6 again) λ ( P + x ) = k i =1 λ ( I i + x ) = k i =1 λ ( I i ) = λ ( P ) . Now let G be an open set in R n . If G = , then G + x = , so clearly λ ( G ) = λ ( G + x ). Otherwise, by the above, a set P is a special polygon such that P G if and only if P + x is a special polygon such that P + x G + x , and then λ ( P ) = λ ( P + x ). Thus, λ ( G ) = sup { λ ( P ) | P is a special polygon such that P G } = sup { λ ( P + x ) | P + x is a special polygon such that P + x G + x } = sup { λ ( Q ) | Q
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