Introduction to Measure Theory

Question 02
What are the properties that must be satisfied for (X, A, μ) to be a probability space?
• A)
  A must be a σ-algebra and μ must be non-negative.
• B)
  A must contain all subsets of X and μ must be finite.
• C)
  A must not be empty and μ must always equal 0.
• D)
  X must be finite and A must be a power set.
Question 03

How does the uniform distribution relate to coin flips?

Question 04
What does it mean for the function S T to converge in probability to S according to the text?
• A)
  It indicates that S T remains constant for all x.
• B)
  It indicates that the frequency of H converges to 1/2.
• C)
  It means S T becomes equal to 1 for all values.
• D)
  It implies S T varies randomly without consistency.
Question 05

What is the Borel σ-algebra denoted by?

Question 06

What are the three properties that define an outer measure?

Question 07
What does Theorem 18 state about the equivalence of measurability types for a set A ⊆ [0, 1]?
• A)
  A is L-measurable if and only if it is C-measurable.
• B)
  A is C-measurable if it can be approximated by open and closed sets.
• C)
  A is C-measurable if its inner measure equals the outer measure.
• D)
  A is L-measurable if it is contained in a closed set.
Question 08

What characterizes a set A ⊆ [0, 1] as L-measurable?

Question 09

What does Theorem 18 imply about the Vitali set V?

Question 10
What is one characterization of measurability for a set A ⊆ [0, 1] according to the text?
• A)
  A set is measurable if it can be expressed as a finite union of intervals.
• B)
  Measurability implies the set can be approximated by closed sets.
• C)
  A set is measurable iff for any ε > 0 it is the difference between a measurable set and an open set.
• D)
  A set A is measurable iff for any ε > 0 there exists a set B ⊆ [0, 1] that is a finite union of intervals and λ ∗ (A ∆ B) < ε.