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Measure Theory Fundamentals

This quiz covers fundamental concepts of Measure Theory including properties of measures, additivity, monotonicity, continuity from above and below, and behavior with null and infinite-measure sets. Requires understanding of measure definitions and theorems.

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Which of the following is not a property of a measure (\mu) on a set (X): (\mu(\emptyset) = 0), (\mu(A \cup B) = \mu(A) + \mu(B)) for disjoint sets (A) and (B), or (\mu(A) \leq \mu(B)) whenever (A \subseteq B)?

  1. \(\mu(\emptyset) = 0\)
  2. \(\mu(A \cup B) = \mu(A) + \mu(B)\) for disjoint sets \(A\) and \(B\)
  3. \(\mu(A) \leq \mu(B)\) whenever \(A \subseteq B\)
Question 2 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A_1, A_2, \ldots) is a sequence of sets in (X) such that (A_n \subseteq A_{n+1}) for all (n), then what is the limit of (\mu(A_n)) as (n) approaches infinity?

  1. \(\mu(\cup_{n=1}^\infty A_n)\)
  2. \(\mu(\cap_{n=1}^\infty A_n)\)
  3. \(\lim_{n\to\infty} \mu(A_n)\)
Question 3 Multiple Choice (Single Answer)

Which of the following is not a type of measure: Lebesgue measure, counting measure, or probability measure?

  1. Lebesgue measure
  2. Counting measure
  3. Probability measure
Question 4 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A) and (B) are sets in (X) such that (A \cap B = \emptyset), then what is the relationship between (\mu(A \cup B)) and (\mu(A) + \mu(B))?

  1. \(\mu(A \cup B) = \mu(A) + \mu(B)\)
  2. \(\mu(A \cup B) < \mu(A) + \mu(B)\)
  3. \(\mu(A \cup B) > \mu(A) + \mu(B)\)
Question 5 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A_1, A_2, \ldots) is a sequence of sets in (X) such that (A_n \cap A_m = \emptyset) for all (n \neq m), then what is the relationship between (\mu(\cup_{n=1}^\infty A_n)) and (\sum_{n=1}^\infty \mu(A_n))?

  1. \(\mu(\cup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu(A_n)\)
  2. \(\mu(\cup_{n=1}^\infty A_n) < \sum_{n=1}^\infty \mu(A_n)\)
  3. \(\mu(\cup_{n=1}^\infty A_n) > \sum_{n=1}^\infty \mu(A_n)\)
Question 6 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A) and (B) are sets in (X) such that (A \subseteq B), then what is the relationship between (\mu(A)) and (\mu(B))?

  1. \(\mu(A) = \mu(B)\)
  2. \(\mu(A) < \mu(B)\)
  3. \(\mu(A) > \mu(B)\)
Question 7 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A) is a set in (X) such that (\mu(A) = 0), then what is the relationship between (\mu(A \cup B)) and (\mu(B)) for any set (B) in (X)?

  1. \(\mu(A \cup B) = \mu(B)\)
  2. \(\mu(A \cup B) < \mu(B)\)
  3. \(\mu(A \cup B) > \mu(B)\)
Question 8 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A_1, A_2, \ldots) is a sequence of sets in (X) such that (A_n \downarrow A) (i.e., (A_1 \supseteq A_2 \supseteq \cdots) and (\cap_{n=1}^\infty A_n = A)), then what is the relationship between (\mu(A_n)) and (\mu(A))?

  1. \(\mu(A_n) = \mu(A)\)
  2. \(\mu(A_n) < \mu(A)\)
  3. \(\mu(A_n) > \mu(A)\)
Question 9 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A_1, A_2, \ldots) is a sequence of sets in (X) such that (A_n \uparrow A) (i.e., (A_1 \subseteq A_2 \subseteq \cdots) and (\cup_{n=1}^\infty A_n = A)), then what is the relationship between (\mu(A_n)) and (\mu(A))?

  1. \(\mu(A_n) = \mu(A)\)
  2. \(\mu(A_n) < \mu(A)\)
  3. \(\mu(A_n) > \mu(A)\)
Question 10 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A) and (B) are sets in (X) such that (\mu(A) = \mu(B)), then what is the relationship between (\mu(A \cup B)) and (\mu(A \cap B))?

  1. \(\mu(A \cup B) = \mu(A \cap B)\)
  2. \(\mu(A \cup B) < \mu(A \cap B)\)
  3. \(\mu(A \cup B) > \mu(A \cap B)\)
Question 11 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A) is a set in (X) such that (\mu(A) = 1), then what is the relationship between (\mu(A \cap B)) and (\mu(B)) for any set (B) in (X)?

  1. \(\mu(A \cap B) = \mu(B)\)
  2. \(\mu(A \cap B) < \mu(B)\)
  3. \(\mu(A \cap B) > \mu(B)\)
Question 12 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A) and (B) are sets in (X) such that (\mu(A) = \mu(B) = \infty), then what is the relationship between (\mu(A \cup B)) and (\mu(A \cap B))?

  1. \(\mu(A \cup B) = \mu(A \cap B)\)
  2. \(\mu(A \cup B) < \mu(A \cap B)\)
  3. \(\mu(A \cup B) > \mu(A \cap B)\)
Question 13 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A_1, A_2, \ldots) is a sequence of sets in (X) such that (\mu(A_n) \to 0) as (n) approaches infinity, then what is the relationship between (\mu(\cup_{n=1}^\infty A_n)) and (0)?

  1. \(\mu(\cup_{n=1}^\infty A_n) = 0\)
  2. \(\mu(\cup_{n=1}^\infty A_n) < 0\)
  3. \(\mu(\cup_{n=1}^\infty A_n) > 0\)
Question 14 Multiple Choice (Single Answer)

Let (\mu) be a measure on a set (X). If (A) and (B) are sets in (X) such that (\mu(A) = \infty) and (\mu(B) = 0), then what is the relationship between (\mu(A \cup B)) and (\mu(A))?

  1. \(\mu(A \cup B) = \mu(A)\)
  2. \(\mu(A \cup B) < \mu(A)\)
  3. \(\mu(A \cup B) > \mu(A)\)
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