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

(Key: correct, incorrect, partially correct.)

- If is the union of a Borel set and a null set, the best one can say about is that it is
- A Borel set.
- A null set.
- A Lebesgue measurable set.
- CORRECT.

- A set.
- A set.
- An arbitrary set.
- A dense set.

- Let be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
- is equal to a set with a null set removed.
- CORRECT.

- is equal to a set with a null set added.
- is contained in a set.
- is equal to a set minus a set of arbitrarily small measure.
- is equal to a set with a set of arbitrarily small measure added.
- contains a set.
- is equal to a set with a null set added and a null set removed.

- is equal to a set with a null set removed.
- Let be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
- is equal to a set with a null set added.
- CORRECT.

- is equal to a set with a null set removed.
- is contained in a set.
- is equal to a set minus a set of arbitrarily small measure.
- is equal to a set with a set of arbitrarily small measure added.
- contains a set.
- is equal to a set with a null set added and a null set removed.

- is equal to a set with a null set added.
- Let be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
- is equal to an open set with a null set added.
- is equal to an open set with a null set removed.
- is contained in an open set.
- is equal to an open set minus a set of arbitrarily small measure.
- CORRECT.

- is equal to an open set with a set of arbitrarily small measure added.
- is equal to a open set with sets of arbitrarily small measure added and removed.
- is equal to a open set with a null set added and a null set removed.

- Let be a Lebesgue measurable set. Of the true statements listed below, which one is the strongest?
- is equal to a closed set with a null set added.
- is equal to a closed set with a null set removed.
- is equal to a closed set with sets of arbitrarily small measure added and removed.
- is equal to a closed set minus a set of arbitrarily small measure.
- is equal to a closed set with a set of arbitrarily small measure added.
- CORRECT.

- contains a closed set.
- is equal to a closed set with a null set added and a null set removed.

- Let be a Lebesgue measurable set of finite measure. Of the true statements listed below, which one is the strongest?
- is equal to a finite union of intervals with a null set added.
- is equal to a finite union of intervals with a null set removed.
- is equal to a finite union of intervals with sets of arbitrarily small measure added and removed.
- CORRECT.

- is equal to a finite union of intervals minus a set of arbitrarily small measure.
- is equal to a finite union of intervals with a set of arbitrarily small measure added.
- CORRECT.

- is contained in a finite union of intervals.
- is equal to a finite union of intervals with a null set added and a null set removed.

- Which of the following classes of sets is
*not*closed under countable unions?- The class of null sets.
- The class of open sets.
- The class of Borel sets.
- The class of Lebesgue measurable sets.
- The class of sets.
- CORRECT.

- The class of sets.
- The class of countable sets.

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