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Measure Theory and Integration

Description: This quiz is designed to assess your understanding of the fundamental concepts and techniques of Measure Theory and Integration.
Number of Questions: 15
Created by:
Tags: measure theory integration measurable sets lebesgue measure integrable functions convergence theorems
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Let (X, Σ, μ) be a measure space. Which of the following sets is not measurable?

  1. A set of real numbers

  2. A set of integers

  3. A set of points in a plane

  4. A set of all rational numbers

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Correct Option: D
Explanation:

The set of all rational numbers is not measurable because it is not a subset of any measurable set.

Which of the following functions is not integrable on the interval [0, 1]?

  1. f(x) = 1/x

  2. f(x) = sin(x)

  3. f(x) = e^x

  4. f(x) = x^2

Show answer
Correct Option: A
Explanation:

The function f(x) = 1/x is not integrable on the interval [0, 1] because it has an infinite discontinuity at x = 0.

Let f be a Lebesgue integrable function on [0, 1]. Which of the following statements is true?

  1. The set of points where f is discontinuous has measure zero.

  2. The set of points where f is continuous has measure one.

  3. The set of points where f is zero has measure zero.

  4. The set of points where f is positive has measure one.

Show answer
Correct Option: A
Explanation:

By the Lebesgue integrability of f, the set of points where f is discontinuous has measure zero.

Which of the following is a property of the Lebesgue measure?

  1. It is translation invariant.

  2. It is countably additive.

  3. It is a complete measure.

  4. All of the above

Show answer
Correct Option: D
Explanation:

The Lebesgue measure is translation invariant, countably additive, and a complete measure.

Let (X, Σ, μ) be a measure space. Which of the following statements is true?

  1. If f and g are measurable functions, then f + g is measurable.

  2. If f and g are measurable functions, then f * g is measurable.

  3. If f is a measurable function and c is a constant, then cf is measurable.

  4. All of the above

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Correct Option: D
Explanation:

All of the above statements are true because the operations of addition, multiplication by a constant, and multiplication of measurable functions are all measurable.

Which of the following is a consequence of the Monotone Convergence Theorem?

  1. If a sequence of measurable functions converges pointwise to a function, then the limit function is measurable.

  2. If a sequence of measurable functions converges in measure to a function, then the limit function is measurable.

  3. If a sequence of measurable functions converges almost everywhere to a function, then the limit function is measurable.

  4. All of the above

Show answer
Correct Option: D
Explanation:

All of the above statements are consequences of the Monotone Convergence Theorem.

Let (X, Σ, μ) be a measure space. Which of the following statements is true?

  1. If f is an integrable function on X, then |f| is also integrable.

  2. If f is an integrable function on X, then f^2 is also integrable.

  3. If f is an integrable function on X, then 1/f is also integrable.

  4. None of the above

Show answer
Correct Option: A
Explanation:

If f is an integrable function on X, then |f| is also integrable because the absolute value function is a measurable function.

Which of the following is a property of the Riemann integral?

  1. It is defined for all continuous functions.

  2. It is defined for all bounded functions.

  3. It is defined for all measurable functions.

  4. None of the above

Show answer
Correct Option: B
Explanation:

The Riemann integral is defined for all bounded functions, but it is not defined for all continuous functions or all measurable functions.

Let f be a function defined on the interval [0, 1]. Which of the following conditions is sufficient to ensure that f is Riemann integrable?

  1. f is continuous on [0, 1].

  2. f is bounded on [0, 1].

  3. f is measurable on [0, 1].

  4. None of the above

Show answer
Correct Option: B
Explanation:

A function f is Riemann integrable if it is bounded on the interval [0, 1].

Which of the following is a consequence of the Dominated Convergence Theorem?

  1. If a sequence of measurable functions converges pointwise to a function, then the limit function is integrable.

  2. If a sequence of measurable functions converges in measure to a function, then the limit function is integrable.

  3. If a sequence of measurable functions converges almost everywhere to a function, then the limit function is integrable.

  4. None of the above

Show answer
Correct Option: A
Explanation:

The Dominated Convergence Theorem states that if a sequence of measurable functions converges pointwise to a function and is dominated by an integrable function, then the limit function is integrable.

Let (X, Σ, μ) be a measure space. Which of the following statements is true?

  1. If f is an integrable function on X, then ∫f dμ = ∫|f| dμ.

  2. If f is an integrable function on X, then ∫f^2 dμ = (∫f dμ)^2.

  3. If f is an integrable function on X, then ∫1/f dμ = 1/∫f dμ.

  4. None of the above

Show answer
Correct Option: D
Explanation:

None of the above statements are true in general.

Which of the following is a property of the Lebesgue integral?

  1. It is linear.

  2. It is monotone.

  3. It is translation invariant.

  4. All of the above

Show answer
Correct Option: D
Explanation:

The Lebesgue integral is linear, monotone, and translation invariant.

Let f be a function defined on the interval [0, 1]. Which of the following conditions is sufficient to ensure that f is Lebesgue integrable?

  1. f is continuous on [0, 1].

  2. f is bounded on [0, 1].

  3. f is measurable on [0, 1].

  4. None of the above

Show answer
Correct Option: C
Explanation:

A function f is Lebesgue integrable if it is measurable on the interval [0, 1].

Which of the following is a consequence of the Fubini-Tonelli Theorem?

  1. If f is an integrable function on a product space X × Y, then ∫∫f(x, y) dx dy = ∫∫f(x, y) dy dx.

  2. If f is an integrable function on a product space X × Y, then ∫∫f(x, y) dx dy ≤ ∫∫f(x, y) dy dx.

  3. If f is an integrable function on a product space X × Y, then ∫∫f(x, y) dx dy ≥ ∫∫f(x, y) dy dx.

  4. None of the above

Show answer
Correct Option: A
Explanation:

The Fubini-Tonelli Theorem states that if f is an integrable function on a product space X × Y, then the iterated integrals ∫∫f(x, y) dx dy and ∫∫f(x, y) dy dx are equal.

Let (X, Σ, μ) be a measure space. Which of the following statements is true?

  1. If f is an integrable function on X, then ∫f dμ = ∫|f| dμ.

  2. If f is an integrable function on X, then ∫f^2 dμ = (∫f dμ)^2.

  3. If f is an integrable function on X, then ∫1/f dμ = 1/∫f dμ.

  4. None of the above

Show answer
Correct Option: D
Explanation:

None of the above statements are true in general.

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