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# mid1 - Write your name clearly on all sheets of paper you...

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Write your name clearly on all sheets of paper you will turn in and possibly number them. Write clearly and large enough to be easily readable. Your poofs must be complete and clearly written. Each question is worth the number of points indicated. Fifty five points will grant you an A but you have to solve at least one between ex. 1 and ex. 2. 1 (15 pts) Let f : R R 2 be a C 1 function ( i.e. f is continuous, differentiable and the derivative is continuous). Prove that μ ( f ( R )) = 0, where μ is the Lebesgue measure on R 2 . 2 Let M 1 and M 2 be two σ -algebras of subsets of X . Let μ 1 be a measure defined on M 1 and μ 2 be a measure on M 2 . a) (10 pts) Give a definition for the measure μ = μ 1 + μ 2 . On which σ -algebra M is μ defined? Show that your definition defines a measure. b) (10 pts) Show that if μ 1 and μ 2 are complete than μ = μ 1 + μ 2 is complete. c) (10 pts) Show that if μ 1 M and μ 2 M are σ -finite than μ = μ 1 + μ 2 is also σ -finite.

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