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# mtsheet1 - Let A be a mble subset of R with m A> 0 Show...

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Measure Theory Fall 2007 Sel¸cuk Demir Exercise Sheet – 1 Exercise 1.1 For a rectangle R in R d , show that | R | = m * ( R ). Exercise 1.2 For a parallellpiped P in R d with vertices 0 ,X 1 ,X 2 , ··· ,X d show that m * ( P ) = | det(( X 1 , ··· ,X d )) | , where ( X 1 , ··· ,X d ) denotes the matrix whose i th row is X i . Exercise 1.3 Show that there exist A,B R d such that A B = but m * ( A B ) 6 = m * ( A ) + m * ( B ). Exercise 1.4
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Unformatted text preview: Let A be a mble subset of R with m ( A ) > 0. Show that (-δ,δ ) ⊂ { x-y : x,y ∈ A } for some δ > 0. Exercise 1.5 If { r n : n ∈ N } is an enumeration of rational numbers, is it necessarily true that ∪ ∞ n =1 ( r n-1 n ,r n + 1 n ) = R ? 1...
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