This quiz is designed to test your knowledge of countably additive measures on a sigma-algebra and related concepts.

Unless otherwise indicated, is a measure space.

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

1. If is a sequence of measurable sets in such that goes to zero, and is a measurable function, what are the most general conditions under which one can assume the integral of on also goes to zero?
   1. One needs to be bounded.
      • CORRECT. This is sufficient, but is rather strong.
   2. One needs to be non-negative.
   3. One needs to be absolutely integrable.
      • CORRECT.
   4. One needs the  to be nested and decreasing.
      • INCORRECT. This is not enough, even with monotone convergence, unless is already absolutely integrable.
   5. One needs to be a simple function.
      • INCORRECT. This is sufficient, but is far too strong.
   6. One needs to be continuous.
      • INCORRECT. This is insufficient, if is unbounded.
   7. One needs to be locally integrable.
      • INCORRECT. This is insufficient, if is unbounded.
2. If  and  are non-negative measures, what does it mean for and to be mutually singular?
   1. Given any measurable set , at most one of and is non-zero.
      • INCORRECT. This can easily fail for most pairs of mutually singular measures.
   2. Given any measurable set , exactly one of and is non-zero.
      • INCORRECT. This can easily fail for most pairs of mutually singular measures.
   3. Every measurable set can be decomposed as where and .
      • INCORRECT. This is a consequence of mutual singularity, but does not quite imply it.
   4. Given any measurable set , at most one of and is finite.
   5. There exists disjoint measurable sets and such that for all measurable .
      • CORRECT.
   6. Every set which is null in is non-null in , and vice versa.
   7. There does not exist a set which is null in and simultaneously.
3. If is a signed measure, how does one define the unsigned measure ?
   1. is equal to the positive variation of mu plus the negative variation of mu.
      • CORRECT.
   2. is equal to if is positive and if is negative.
      • INCORRECT. Most signed measures are neither positive nor negative, but need to be decomposed into components.
   3. is the measure which gives each set a measure of .
   4. is the measure with density with respect to Lebesgue measure .
      • PARTIALLY. This is only true when is absolutely continuous with respect to Lebesgue measure.
   5. is the positive variation of .
   6. is the difference of the positive and negative variations of .
   7. is infinity if is an infinite measure, and is equal to when is a finite measure.
      • INCORRECT. Measures are not numbers.
4. Let be an absolutely integrable complex-valued function with respect to a measure of . What is the most precise relationship between the integral of and the integral of ?
   1. The integral of has magnitude less than or equal to the integral of .
      • CORRECT.
   2. The magnitude of the integral of is equal to the integral of .
   3. The integral of has magnitude greater than or equal to the integral of .
   4. The real and imaginary parts of the integral of are less than the integral of .
      • INCORRECT. This is true but is not the most precise statement one can make.
   5. The sum of squares of the real and imaginary parts of the integral of add up to the square of the integral of .
   6. Nothing can be said unless has a fixed sign or phase.
   7. The integral of is strictly smaller in magnitude than the integral of .
   8. The integral of lies between the integral of and the negative integral of .
      • INCORRECT. This does not make sense since the integral of is complex.
5. What does it mean for a signed measure to be supported on a measurable set ?
   1. whenever is disjoint from and measurable.
      • CORRECT.
   2. .
   3. .
      • PARTIALLY. This would only be correct if mu was unsigned.
   4. is the smallest set such that .
   5. is the largest set such that is non-zero.
   6. is non-zero.
   7. is non-zero whenever is contained in and measurable.
6. Let be a Lebesgue measurable subset of the real line. Which of the following statements is true?
   1. If is bounded, then it has finite Lebesgue measure.
      • CORRECT.
   2. If has finite Lebesgue measure, then it is bounded.
   3. If is bounded, then it is finite.
   4. If is unbounded, then it has infinite Lebesgue measure.
   5. If has zero measure, then it is bounded.
   6. If is uncountable, it has non-zero measure.
   7. If is uncountable, then it is unbounded.
7. The support of a finite measure is
   1. The largest set which does not contain null sets.
   2. The smallest set whose complement is null.
   3. The smallest set whose complement is totally null.
   4. Not unique; a measure can have more than one support.
      • CORRECT.
   5. The union of all the sets of positive measure.
   6. The intersection of all the sets of positive measure.
   7. The intersection of all the sets of full measure.
8. Let be an unsigned measure, and be a signed measure. Which of the following statements is true?
   1. If , then is absolutely continuous with respect to .
      • CORRECT.
   2. If , then is absolutely continuous with respect to nu.
   3. If , then is absolutely continuous with respect to .
   4. If is absolutely continuous with respect to . then .
   5. If is absolutely continuous with respect to , then .
   6. If , then is absolutely continuous with respect to .
   7. If is absolutely continuous with respect to , then we have for some constant .
9. Let be an unsigned measure, and be a signed measure. What does it mean for nu to be absolutely continuous with respect to ?
   1. Every point has a -measure of zero.
      • INCORRECT. This is what it means for to be continuous, not absolutely continuous.
   2. The Radon-Nikodym derivative is a continuous function.
   3. The Radon-Nikodym derivative is a continuous function almost everywhere.
   4. Every set which has a -measure of zero also has a -measure of zero.
      • CORRECT.
   5. Every set which has a -measure of zero also has a -measure of zero.
   6. For any measurable set , .
   7. is supported on a set of positive -measure.
   8. is not absolutely singular with respect to .

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