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\textbf{Difficult MCQs on Measure Theory and Integration with Answers}
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\[
\textbf{Q1: Which of the following is a property of a measure on a measurable space?}
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\[
\text{(A) } \text{It is a set function that is countably additive.}
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\[
\text{(B) } \text{It is a function that maps sets to real numbers.}
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\[
\text{(C) } \text{It must be a finite function.}
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\[
\text{(D) } \text{It is always continuous.}
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\[
\textbf{Answer: (A) It is a set function that is countably additive.}
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\textbf{Q2: What is the definition of the Lebesgue integral?}
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\[
\text{(A) } \text{It is the integral of a function with respect to a measure.}
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\[
\text{(B) } \text{It is the sum of the function over a set.}
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\[
\text{(C) } \text{It is defined only for continuous functions.}
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\[
\text{(D) } \text{It is always finite for all measurable functions.}
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\[
\textbf{Answer: (A) It is the integral of a function with respect to a measure.}
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\textbf{Q3: Which of the following is true about the Dominated Convergence Theorem?}
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\[
\text{(A) } \text{It allows for the interchange of limit and integral under certain conditions.}
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\[
\text{(B) } \text{It applies only to bounded functions.}
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\[
\text{(C) } \text{It requires the integrand to be continuous.}
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\[
\text{(D) } \text{It only applies to finite measure spaces.}
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\[
\textbf{Answer: (A) It allows for the interchange of limit and integral under certain conditions.}
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\textbf{Q4: What is the main difference between the Lebesgue integral and the Riemann integral?}
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\[
\text{(A) } \text{The Lebesgue integral deals with measurable functions, while the Riemann integral deals with continuous functions.}
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\[
\text{(B) } \text{The Lebesgue integral sums over sets of points, while the Riemann integral sums over intervals.}
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\[
\text{(C) } \text{The Lebesgue integral always converges, while the Riemann integral never does.}
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\[
\text{(D) } \text{The Riemann integral is only defined for bounded functions.}
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\[
\textbf{Answer: (B) The Lebesgue integral sums over sets of points, while the Riemann integral sums over intervals.}
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\textbf{Q5: If a sequence of measurable functions \(f_n\) converges pointwise to a function \(f\), under what condition does the Dominated Convergence Theorem allow the interchange of the limit and the integral?}
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\[
\text{(A) } \text{If there is an integrable function \(g\) such that \(|f_n| \leq g\) for all \(n\).}
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\[
\text{(B) } \text{If the sequence \(f_n\) is bounded.}
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\[
\text{(C) } \text{If the sequence \(f_n\) is continuous.}
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\[
\text{(D) } \text{If \(f\) is continuous.}
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\[
\textbf{Answer: (A) If there is an integrable function \(g\) such that \(|f_n| \leq g\) for all \(n\).}
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\textbf{Q6: Which of the following is true about the Carathéodory criterion for a measure?}
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\[
\text{(A) } \text{A set is measurable if it can be approximated from the outside by open sets and from the inside by closed sets.}
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\[
\text{(B) } \text{A set is measurable if the outer measure of the set equals the sum of the outer measures of disjoint parts.}
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\[
\text{(C) } \text{A set is measurable if it is contained in a Borel set.}
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\[
\text{(D) } \text{A set is measurable if it is finite.}
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\[
\textbf{Answer: (B) A set is measurable if the outer measure of the set equals the sum of the outer measures of disjoint parts.}
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\textbf{Q7: Which of the following is true about the outer measure of a set?}
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\[
\text{(A) } \text{The outer measure of any set is always finite.}
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\[
\text{(B) } \text{The outer measure of a set is always less than or equal to the measure of the set.}
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\[
\text{(C) } \text{The outer measure of a set is always greater than or equal to the measure of the set.}
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\[
\text{(D) } \text{The outer measure of a set is independent of the chosen covering.}
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\[
\textbf{Answer: (B) The outer measure of a set is always less than or equal to the measure of the set.}
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\[
\textbf{Q8: What is the purpose of the Fubini-Tonelli theorem in measure theory?}
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\[
\text{(A) } \text{It allows the interchange of the order of integration for double integrals.}
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\[
\text{(B) } \text{It allows the calculation of the measure of a product space.}
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\[
\text{(C) } \text{It proves that integrable functions are always continuous.}
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\[
\text{(D) } \text{It gives a condition for the convergence of improper integrals.}
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\[
\textbf{Answer: (A) It allows the interchange of the order of integration for double integrals.}
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\[
\textbf{Q9: In the context of measure theory, what is the difference between an absolutely continuous measure and a singular measure?}
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\[
\text{(A) } \text{An absolutely continuous measure has a density with respect to Lebesgue measure, while a singular measure does not.}
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\[
\text{(B) } \text{An absolutely continuous measure is finite, while a singular measure is infinite.}
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\[
\text{(C) } \text{An absolutely continuous measure is always discrete, while a singular measure is continuous.}
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\[
\text{(D) } \text{An absolutely continuous measure is atomic, while a singular measure is nonatomic.}
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\[
\textbf{Answer: (A) An absolutely continuous measure has a density with respect to Lebesgue measure, while a singular measure does not.}
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\[
\textbf{Q10: What is the support of a measure \(\mu\)?}
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\[
\text{(A) } \text{The closure of the set of points where \(\mu\) is positive.}
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\[
\text{(B) } \text{The set where \(\mu\) is finite.}
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\[
\text{(C) } \text{The set where \(\mu\) is zero.}
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\[
\text{(D) } \text{The set where \(\mu\) is nonzero.}
\]
\[
\textbf{Answer: (A) The closure of the set of points where \(\mu\) is positive.}
\]