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### hw4

Course: MATH 136, Spring 2008
School: Tufts
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Word Count: 503

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University Tufts Department of Mathematics Math 136 Homework 4 Due, Tuesday, March 4, 2008 in class. Here is a theorem that may be used on this homework even if we don't prove it until after this homework is due. Theorem 1 Let A Rn . Then A has measure zero if and only if for each collection of open rectangles (with sides parallel the axes) {Uj j N}so that &gt; 0, there is a countable A jN Uj and j=1...

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University Tufts Department of Mathematics Math 136 Homework 4 Due, Tuesday, March 4, 2008 in class. Here is a theorem that may be used on this homework even if we don't prove it until after this homework is due. Theorem 1 Let A Rn . Then A has measure zero if and only if for each collection of open rectangles (with sides parallel the axes) {Uj j N}so that > 0, there is a countable A jN Uj and j=1 v(Uj ) < . 1. (20 points) You will show that open rectangles have volume and the volume of an open rectangle is the same as the volume of its closure, a closed rectangle. (a) Let a R and b R with a < b. Show using the definition of integral and a good partition that 1[a,b] = b - a. This shows that the volume of [a, b] as a set is the same as the volume of [a, b] as an interval. (b) Let a R and b R with a < b. Prove v((a, b)) = v([a, b]) using the limit theorem for the integral and the partitions Pn = [a, a + 1/n], [a + 1/n, b - 1/n], [b - 1/n, b] for n > 2/(b - a). Note that this partition is especially fine near where 1(a,b) is discontinuous. (c) Let aj and bj be points in R with aj < bj for j = 1, 2. Prove that the open rectangle (a1 , b1 ) (a2 , b2 ) has the same volume as the closed rectangle [a1 , b1 ] [a2 , b2 ]. 2. (25 points) (a) Prove that if A is a bounded in set R with volume zero, then A has measure zero. (b) Prove that any subset of a set of measure zero also has measure zero. (c) Let A Rn be compact. Prove that if A has measure zero then A has volume zero. 3. (15 points) Let A Rn be bounded and let R0 Rn be a rectangle with sides parallel the coordinate axes and A R0 . Let f : A R be bounded. Let P be a partition of R0 . Explain why L(f, P ) any Riemann sum for f and P U (f, P ). 4. (15 points) Let A Rn be bounded and let g : A R be bounded. Let S0 be a rectangle with sides parallel the axes that contains A. Let > 0 and let P be a partition of S0 . Assume R P, x1 R, x2 R, |g(x1 ) - g(x2 )| < . Prove that U (g, P )-L(g, P ) v(S0 ). (This proof is like one part of the proof of Lebesgue's Theorem.) 5. (10 points) Use Lebesgue's Theorem to discuss integrability of the following functions (a) f : [0, 1] R defined by f (x) = 1 x+2 + tan(ex-1 ) sin(1/x) x = 0 . Use Lebesgue's theorem to prove 0 x=0 (b) Let f : [-1, -1] R be defined by f (x) = f is integrable on [-1, 1]. 6. (15 points) Let f (x) = 0 sin(1/x) if x = 0 if x = 0 (a) Find the oscillation O(f, 0) and prove your answer using the definition of oscillation. (b) Find the oscillation O(f, 1) and prove your answer using the definition of oscillation. 1
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