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Unformatted text preview: 18.100B/C: Fall 2008 Homework 10 Available Wednesday, November 12 Due Wednesday, November 19 1. Recall that a subset N ⊆ R is said to have measure if, for each ǫ > , there is a sequence (finite or countable) of balls ( B n ) with radii r n so that N ⊆ uniontext n B n and ∑ n r n < ǫ . (a) (10pts) Let N be a set of measure in R . Prove that the complement N c is dense in R . (b) (5pts) Show (from the definition) that the only open set that has measure is ∅ . (c) (5 pts) Can measure sets be closed? Non-compact? Dense? 2. (20pts) Consider the following two real-valued functions on [0 , 1] . f ( x ) = braceleftBigg 1 , x = 1 n for some n ∈ N , otherwise , g ( x ) = braceleftBigg n, x = 1 n for some n ∈ N , otherwise . Show (from the definition) that f ∈ R (i.e. f is Riemann-integrable), with integraltext 1 f ( x ) dx = , but g / ∈ R . 3. Problem 3, page 138 in Rudin . 4. Problem 17, page 141 in Rudin ....
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This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.
- Fall '10