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Quiz: Countably additive measures

Page history last edited by RH 11 years, 9 months ago

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

 

Unless otherwise indicated, Formula is a measure space.

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

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

 

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