Quiz: Measurable subsets of the real line

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

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

Discuss this quiz

(Key: correct, incorrect, partially correct.)

1. If is the union of a Borel set and a null set, the best one can say about is that it is
1. A Borel set.
2. A null set.
3. A Lebesgue measurable set.
• CORRECT.
4. A set.
5. A set.
6. An arbitrary set.
7. A dense set.
2. Let be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
1. is equal to a set with a null set removed.
• CORRECT.
2. is equal to a set with a null set added.
3. is contained in a set.
4. is equal to a set minus a set of arbitrarily small measure.
5. is equal to a set with a set of arbitrarily small measure added.
6. contains a set.
7. is equal to a set with a null set added and a null set removed.
3. Let be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
1. is equal to a set with a null set added.
• CORRECT.
2. is equal to a set with a null set removed.
3. is contained in a set.
4. is equal to a set minus a set of arbitrarily small measure.
5. is equal to a set with a set of arbitrarily small measure added.
6. contains a set.
7. is equal to a set with a null set added and a null set removed.
4. Let be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
1. is equal to an open set with a null set added.
2. is equal to an open set with a null set removed.
3. is contained in an open set.
4. is equal to an open set minus a set of arbitrarily small measure.
• CORRECT.
5. is equal to an open set with a set of arbitrarily small measure added.
6. is equal to a open set with sets of arbitrarily small measure added and removed.
7. is equal to a open set with a null set added and a null set removed.
5. Let be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
1. is equal to a closed set with a null set added.
2. is equal to a closed set with a null set removed.
3. is equal to a closed set with sets of arbitrarily small measure added and removed.
4. is equal to a closed set minus a set of arbitrarily small measure.
5. is equal to a closed set with a set of arbitrarily small measure added.
• CORRECT.
6. contains a closed set.
7. is equal to a closed set with a null set added and a null set removed.
6. Let be a Lebesgue measurable set of finite measure. Of the true statements listed below, which one is the strongest?
1. is equal to a finite union of intervals with a null set added.
2. is equal to a finite union of intervals with a null set removed.
3. is equal to a finite union of intervals with sets of arbitrarily small measure added and removed.
• CORRECT.
4. is equal to a finite union of intervals minus a set of arbitrarily small measure.
5. is equal to a finite union of intervals with a set of arbitrarily small measure added.
• CORRECT.
6. is contained in a finite union of intervals.
7. 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 sets.
• CORRECT.
6. The class of sets.
7. The class of countable sets.