1873_solutions

Deduce that the set x u fx 0 is countable 16

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Unformatted text preview: ons 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 and since 0, 1 does not have measure zero, we deduce from the theorem on unions that the set 0, 1 Q does not have measure zero. 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  An n1 Ý  Bj j1 has measure zero. We see similarly that the set Ý  Bj j1 Ý  An n1 has measure zero. Now we consider intersections. Since Ý Ý n1 j1 Ý 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. Suppose that A has measure zero and that  0. Choose an expanding sequence E n of elementary sets such that m E n  /2 for every n and such that 393 Ý  En. A n1 For each n, choose an open elementary set U n that includes E n such that m U n  . 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: a. Choose an expanding sequence E n of elementary sets such that m E n  /2 for every n and such that Ý A  En. n1 For each n, choose an open elementary set U n such that E n U n and such that m U n  m E n  n 1 . 2 b. For each n define n Vn   Uj, j1 observe that Vn  Un Þ U1 E1 Þ U2 and then deduce that m V n  for each n. E2 Þ  Þ Un 1 En 1 Some Exercises on Integrability 1. If S is a set of real numbers then the boundary of S is defined to be the set S R S. Prove that if S is a subset of an interval a, b then the function  S is Riemann integrable on a, b if and only if the boundary of S has measure zero. The function  S fails to be continuous at a number x if and only if x belongs to the boundary of S....
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