1873_solutions

2 2 sin 2 1 cos 2 1 d 0 this exercise is

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Unformatted text preview: ad the set theory, must assume the existence of the Sierpinski example for the purposes of this exercise. Using Sierpinski’s example, prove that the statement of Fichtenholz’s theorem becomes false if we widen it to include functions that are almost Riemann integrable. Exercises 5 and 6 are offered as a resource for a special project. Alternative 17 Sets of Measure Zero 396 Some Exercises on the Measure Zero Concept 1. Prove that an elementary set E has measure zero if and only if m E  0. Suppose that E is an elementary set. In the event that m E  0, it is clear that E has (nineteenth century) measure zero. On the other hand, if m E  0 then E includes a closed bounded interval of positive length and it follows from the discussion on closed bounded sets of measure zero that E can’t have measure zero. 2. Prove that every countable set has measure zero. This fact follows at once from the theorem on unions of sets of measure zero the fact that a singleton has measure zero. 3. Give an example of an uncountable closed bounded set that has measure zero. We have already observed that the Cantor set has measure zero. 4. Prove that if U is a nonempty open set then U has a closed bounded subset H that does not have measure zero. A nonempty open set must include a closed bounded interval with positive length. 5. Prove that the set of all irrational numbers in the interval 0, 1 does not have measure zero. The set 0, 1 Q, being countable, must have measure zero. Since 0, 1  QÞ 0, 1 0, 1 Q 6. For the purposes of this exercise we agree to call two sets A and B almost equal to each other if both of the sets A B and B A have measure zero. Prove that if A n and B n are sequences of sets and if A n and B n are almost equal to each other then the sets Ý Ý  An  Bn and n1 n1 are almost equal to each other. Can the same assertion be made for intersections? We observe that Ý Ý Ý n1 j1 Ý  An  Bj    An Now, given any n, since the set A n Bj. n1 j1 B j has measure zero for all j, we see at once that the set Ý  An Bj j1 has measure zero. It follows from the theorem on unions that the set Ý Ý n1 j1  An  Bj has measure zero. We see similarly that the set Ý Ý j1 n1 Ý Ý Ý n1 j1  Bj  An has measure zero. Now we consider intersections. Since Ý  An  Bj    An Bj n1 j1 we see easily that this set too has measure zero. 7. Prove that if A has measure zero and if  0 then there exists a sequence U n of open elementary sets whose union includes A and for which m U n  for every n. 397 8. Prove that if A has measure zero and if  0 then there exists an expanding sequence U n of open elementary sets whose union includes A and for which m U n  for every n. Hint: Suppose that A has measure zero and that  0. Follow the step by step procedure outlined below: 1. a. Choose an expanding sequence E n of elementary sets such that m E n  /2 for every n and such that Ý  En. A n1 For each n, choose an open elementary set U n such that E n U...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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