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Quiz: Measurable subsets of the real line

Page history last edited by RH 11 years, 11 months ago

This quiz is designed to test your knowledge of the sigma-algebra of Lebesgue measurable sets of Formula and related concepts.

 

All sets are subsets of the real line Formula unless otherwise indicated.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. If Formula is the union of a Borel set and a null set, the best one can say about Formula is that it is
    1. A Borel set.
    2. A null set.
    3. A Lebesgue measurable set.
      • CORRECT.
    4. A Formula set.
    5. A Formula set.
    6. An arbitrary set.
    7. A dense set.
  2. Let Formula be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
    1. Formula is equal to a Formula set with a null set removed.
      • CORRECT.
    2. Formula is equal to a Formula set with a null set added.
    3. Formula is contained in a Formula set.
    4. Formula is equal to a Formula set minus a set of arbitrarily small measure.
    5. Formula is equal to a Formula set with a set of arbitrarily small measure added.
    6. Formula contains a Formula set.
    7. Formula is equal to a Formula set with a null set added and a null set removed.
  3. Let Formula be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
    1. Formula is equal to a Formula set with a null set added.
      • CORRECT.
    2. Formula is equal to a Formula set with a null set removed.
    3. Formula is contained in a Formula set.
    4. Formula is equal to a Formula set minus a set of arbitrarily small measure.
    5. Formula is equal to a Formula set with a set of arbitrarily small measure added.
    6. Formula contains a Formula set.
    7. Formula is equal to a Formula set with a null set added and a null set removed.
  4. Let Formula be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
    1. Formula is equal to an open set with a null set added.
    2. Formula is equal to an open set with a null set removed.
    3. Formula is contained in an open set.
    4. Formula is equal to an open set minus a set of arbitrarily small measure.
      • CORRECT.
    5. Formula is equal to an open set with a set of arbitrarily small measure added.
    6. Formula is equal to a open set with sets of arbitrarily small measure added and removed.
    7. Formula is equal to a open set with a null set added and a null set removed.
  5. Let Formula be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
    1. Formula is equal to a closed set with a null set added.
    2. Formula is equal to a closed set with a null set removed.
    3. Formula is equal to a closed set with sets of arbitrarily small measure added and removed.
    4. Formula is equal to a closed set minus a set of arbitrarily small measure.
    5. Formula is equal to a closed set with a set of arbitrarily small measure added.
      • CORRECT.
    6. Formula contains a closed set.
    7. Formula is equal to a closed set with a null set added and a null set removed.
  6. Let Formula be a Lebesgue measurable set of finite measure. Of the true statements listed below, which one is the strongest?
    1. Formula is equal to a finite union of intervals with a null set added.
    2. Formula is equal to a finite union of intervals with a null set removed.
    3. Formula is equal to a finite union of intervals with sets of arbitrarily small measure added and removed.
      • CORRECT.
    4. Formula is equal to a finite union of intervals minus a set of arbitrarily small measure.
    5. Formula is equal to a finite union of intervals with a set of arbitrarily small measure added.
      • CORRECT.
    6. Formula is contained in a finite union of intervals.
    7. Formula is equal to a finite union of intervals with a null set added and a null set removed.
  7. Which of the following classes of sets is not closed under countable unions?
    1. The class of null sets.
    2. The class of open sets.
    3. The class of Borel sets.
    4. The class of Lebesgue measurable sets.
    5. The class of Formula sets.
      • CORRECT.
    6. The class of Formula sets.
    7. The class of countable sets.

 

Score:  

 

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