TRB PG Assistant - Lebesgue measure and integral on R - integrals of bounded measurable functions, comparison of Riemann and Lebesgue integrals, monotone convergence theorem, repeated integrals Online Test (MCQ Quiz)

1. What is the definition of Lebesgue outer measure of a set E in R?

A) The infimum of the sum of lengths of open intervals covering E. B) The supremum of the sum of lengths of open intervals covering E. C) The infimum of the sum of lengths of closed intervals covering E. D) The supremum of the sum of lengths of closed intervals covering E.

2. Which property does the Lebesgue outer measure always satisfy?

A) Countable additivity B) Subadditivity C) Finite additivity D) Monotonicity only

3. A set E in R is Lebesgue measurable if and only if for every set A in R, the following equality holds:

A) m*(A) = m*(A ∩ E) + m*(A ∩ E^c) B) m*(A) = m*(A ∪ E) + m*(A ∩ E^c) C) m*(A) = m*(A ∩ E) + m*(A ∪ E^c) D) m*(A) = m*(A ∪ E) + m*(A ∪ E^c)

4. What is the Lebesgue measure of a single point in R?

A) 1 B) 0 C) Infinity D) Undefined

5. The Lebesgue measure of the set of rational numbers Q in R is:

A) 1 B) Infinity C) 0 D) Undefined

6. Which of the following sets is Lebesgue measurable?

A) A non-measurable set B) The Cantor set C) A Vitali set D) The set of all points in R that are not in any open interval of length 1/n for any n in N.

7. Let f be a bounded measurable function on a measurable set E with |f(x)| <= M for all x in E. The Lebesgue integral of f over E, denoted by ∫_E f dm, is defined as:

A) The limit of Riemann sums as the partition becomes finer. B) The supremum of integrals of simple functions less than or equal to f. C) The infimum of integrals of simple functions greater than or equal to f. D) The integral of the function over the set E using the Lebesgue measure.

8. If f is a Riemann integrable function on [a, b], then f is also Lebesgue integrable on [a, b], and their integrals are equal. This statement is:

A) True B) False C) True only if f is continuous D) True only if f is bounded

9. Consider a function f that is zero everywhere except at a single point c, where f(c) = 1. Is f Riemann integrable on [a, b] containing c?

A) Yes, the integral is 0. B) No, because the set of discontinuities has positive measure. C) Yes, the integral is 1. D) No, because the function is not continuous.

10. What is the relationship between Riemann integrability and Lebesgue integrability for a bounded function on a closed interval?

A) Lebesgue integrability implies Riemann integrability. B) Riemann integrability implies Lebesgue integrability. C) They are equivalent for all bounded functions. D) They are equivalent only for continuous functions.

11. The Monotone Convergence Theorem (MCT) applies to a sequence of:

A) Arbitrary measurable functions. B) Non-negative measurable functions. C) Bounded measurable functions. D) Continuously differentiable functions.

12. If {f_n} is a sequence of non-negative measurable functions such that f_n(x) ≤ f_{n+1}(x) for all n and x, and f(x) = lim_{n→∞} f_n(x), then the MCT states:

A) ∫ f dm = lim_{n→∞} ∫ f_n dm B) ∫ f dm ≤ lim_{n→∞} ∫ f_n dm C) ∫ f dm ≥ lim_{n→∞} ∫ f_n dm D) ∫ f dm = sup_{n} ∫ f_n dm

13. The MCT is crucial for proving the integrability of functions that are defined as:

A) Limits of sequences of simple functions. B) Limits of sequences of step functions. C) Limits of sequences of continuous functions. D) Limits of sequences of bounded functions.

14. Let f_n(x) = 1/n for all x in [0, 1]. What is lim_{n→∞} ∫_[0,1] f_n dm?

A) 1 B) 0 C) Infinity D) Undefined

15. Let f_n(x) = x^n on [0, 1]. What is lim_{n→∞} ∫_[0,1] f_n dm?

A) 1 B) 0 C) 1/2 D) Undefined

16. The Dominated Convergence Theorem (DCT) requires that the sequence of measurable functions {f_n} be dominated by an integrable function g, meaning:

A) |f_n(x)| ≤ g(x) for all n and x B) f_n(x) ≤ g(x) for all n and x C) |f_n(x)| ≥ g(x) for all n and x D) f_n(x) ≥ g(x) for all n and x

17. If f_n → f pointwise and |f_n| ≤ g where g is integrable, the DCT states:

A) ∫ f dm = lim_{n→∞} ∫ f_n dm B) ∫ f dm ≤ lim_{n→∞} ∫ f_n dm C) ∫ f dm ≥ lim_{n→∞} ∫ f_n dm D) ∫ f dm = sup_{n} ∫ f_n dm

18. The DCT is more general than the MCT because it allows for:

A) Non-monotonic sequences of functions. B) Sequences that are not necessarily non-negative. C) Functions that are not necessarily bounded. D) All of the above.

19. What is the primary difference between repeated integrals (or iterated integrals) and multiple integrals in the context of Lebesgue integration?

A) Repeated integrals are always equal to multiple integrals. B) Repeated integrals are computed by integrating with respect to one variable at a time, while multiple integrals are computed over a region. C) Multiple integrals are only defined for continuous functions. D) Repeated integrals are only defined for functions of one variable.

20. Fubini's Theorem relates multiple integrals to repeated integrals under certain conditions. What is a key condition for Fubini's Theorem to hold?

A) The function must be continuous. B) The function must be non-negative. C) The function must be absolutely integrable. D) The domain must be a rectangle.

21. If f(x, y) is non-negative and measurable on R^2, then Fubini's Theorem guarantees:

A) ∫∫ f(x,y) dx dy = ∫ [∫ f(x,y) dy] dx B) ∫∫ f(x,y) dx dy ≠ ∫ [∫ f(x,y) dy] dx C) ∫∫ f(x,y) dx dy = ∫ [∫ f(x,y) dx] dy D) ∫∫ f(x,y) dx dy = sup [∫ [∫ f(x,y) dy] dx]

22. What is Tonelli's Theorem?

A) It states that if f is absolutely integrable, then the repeated integrals equal the double integral. B) It states that for a non-negative measurable function, the repeated integrals are equal and equal to the double integral. C) It is a special case of Fubini's Theorem for continuous functions. D) It provides conditions for interchanging order of integration for Riemann integrals.

23. If ∫∫ |f(x,y)| dx dy < ∞, then which of the following is true?

A) ∫∫ f(x,y) dx dy = ∫ [∫ f(x,y) dy] dx = ∫ [∫ f(x,y) dx] dy B) ∫∫ f(x,y) dx dy ≠ ∫ [∫ f(x,y) dy] dx C) ∫∫ f(x,y) dx dy can be different from ∫ [∫ f(x,y) dy] dx D) The repeated integrals may not exist.

24. Consider f(x, y) = (x^2 - y^2) / (x^2 + y^2)^2 for (x, y) ≠ (0, 0). If we calculate the repeated integrals over the unit square [0, 1] x [0, 1], what do we find?

A) Both repeated integrals are 0 and equal to the double integral. B) The repeated integrals exist and are equal, but the double integral does not exist. C) The repeated integrals exist and are equal, and the double integral is also equal to them. D) The repeated integrals exist and are equal, but the double integral is finite and different.

25. The concept of 'bounded measurable function' is important for the initial definition of the Lebesgue integral because:

A) All functions are bounded and measurable. B) It simplifies the definition by relating it to the measure of sets. C) It ensures the function is Riemann integrable. D) It guarantees the function is continuous.

26. For a bounded measurable function f on a measurable set E, the Lebesgue integral ∫_E f dm is defined as the limit of integrals of simple functions that approximate f. Specifically, it's the supremum of integrals of simple functions s such that 0 ≤ s ≤ f.

A) True, for f ≥ 0. B) True, for all bounded measurable f. C) False, it's the infimum of integrals of simple functions s such that s ≥ f. D) False, it's defined using Riemann sums.

27. If f is Riemann integrable on [a, b], the set of discontinuities of f has Lebesgue measure zero. This is a key reason why the Lebesgue integral coincides with the Riemann integral for such functions.

A) True B) False C) True only if f is continuous D) False, the set of discontinuities can have positive measure.

28. Which theorem provides the foundation for interchanging the order of integration in Lebesgue calculus?

A) Monotone Convergence Theorem B) Dominated Convergence Theorem C) Fubini's Theorem D) Fatou's Lemma

29. Let f_n(x) = 1/n * sin(nx) on [0, 1]. What is lim_{n→∞} ∫_[0,1] f_n dm?

A) 1 B) 0 C) Undefined D) Infinity

30. The Lebesgue integral ∫_E f dm is defined for non-negative measurable functions f as the supremum of ∫_E s dm over all simple functions s such that 0 ≤ s ≤ f.

A) True B) False C) True only if f is bounded D) False, it's the infimum.

31. What is Fatou's Lemma?

A) It states that the integral of a limit is the limit of the integrals for a dominated sequence. B) It states that for a sequence of non-negative measurable functions {f_n}, ∫ lim inf f_n dm ≤ lim inf ∫ f_n dm. C) It states that for a sequence of non-positive measurable functions {f_n}, ∫ lim sup f_n dm ≥ lim sup ∫ f_n dm. D) It provides conditions for interchanging order of integration.

32. Fatou's Lemma is useful for establishing the existence of integrals for sequences of functions where:

A) The functions are bounded and converge uniformly. B) The functions are non-negative and converge pointwise. C) The functions are dominated by an integrable function. D) The functions are continuous and their limit is continuous.

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

A) The integral of a sum is the sum of the integrals. B) The integral of a limit of a sequence of non-negative functions is the limit of their integrals. C) The integral of a function over a union of disjoint sets is the sum of the integrals over those sets. D) The integral of a bounded function over a set of measure zero is zero.

34. If f is a bounded measurable function on R, and E is a measurable set with m(E) = 0, then ∫_E f dm is:

A) Undefined B) Infinity C) 0 D) 1

35. The Lebesgue integral is defined for a wider class of functions than the Riemann integral. Which type of function, problematic for Riemann integration, is handled well by Lebesgue integration?

A) Continuous functions B) Functions with a finite number of discontinuities C) Functions with discontinuities on a set of measure zero D) Functions with discontinuities on a set of positive measure

36. Let f(x) = 1 if x is rational in [0, 1] and f(x) = 0 if x is irrational in [0, 1]. Is f Riemann integrable on [0, 1]?

A) Yes, the integral is 0. B) Yes, the integral is 1. C) No, because the set of discontinuities has positive measure. D) No, because the function is not continuous.

37. What is the Lebesgue integral of the Dirichlet function f(x) = 1_{Q}(x) (indicator of rationals) over [0, 1]?

A) 1 B) 0 C) 1/2 D) Undefined

38. The definition of the Lebesgue integral for a general measurable function f is to split it into its positive and negative parts, f = f^+ - f^-, where f^+(x) = max(f(x), 0) and f^-(x) = max(-f(x), 0). The integral is then defined as:

A) ∫ f^+ dm + ∫ f^- dm B) ∫ f^+ dm - ∫ f^- dm C) ∫ f^- dm - ∫ f^+ dm D) max(∫ f^+ dm, ∫ f^- dm)

39. A function f is Lebesgue integrable if and only if:

A) It is continuous and bounded. B) It is Riemann integrable. C) Both ∫ f^+ dm and ∫ f^- dm are finite. D) ∫ |f| dm is finite.

40. The condition ∫ |f| dm < ∞ for a measurable function f implies that:

A) f is Lebesgue integrable. B) f is Riemann integrable. C) f is continuous. D) f is bounded.

41. Which of the following is NOT a property of Lebesgue measure?

A) Non-negativity B) Monotonicity C) Countable additivity D) Finite subadditivity

42. If E is a measurable set and A is any subset of E, then m(A) ≤ m(E). This property is called:

A) Additivity B) Monotonicity C) Invariance under translation D) Completeness

43. The Lebesgue integral of a constant function c over a measurable set E is:

A) c B) m(E) C) c * m(E) D) 0

44. The concept of 'measurable function' is fundamental to Lebesgue integration. A function f is measurable if:

A) It is continuous. B) Its inverse image of any open set is measurable. C) Its inverse image of any measurable set is measurable. D) It is bounded.

45. If f_n → f pointwise and f_n are non-negative, the MCT implies ∫ f dm ≤ lim inf ∫ f_n dm. This is:

A) True B) False C) True only if f_n are bounded D) False, it should be '='.

46. If we have a sequence of measurable functions f_n that converges to f in L^1 norm (i.e., ∫ |f_n - f| dm → 0), then it implies:

A) f_n → f pointwise B) f_n → f uniformly C) lim ∫ f_n dm = ∫ f dm D) None of the above

47. Consider the function f(x, y) = 1 on the unit square [0, 1] x [0, 1]. Calculate the repeated integral ∫_0^1 (∫_0^1 1 dy) dx.

A) 0 B) 1 C) 2 D) Undefined

48. If f is Riemann integrable on [a, b], then f is Lebesgue integrable on [a, b]. What about the converse?

A) The converse is always true. B) The converse is true if f is continuous. C) The converse is true if the set of discontinuities of f has measure zero. D) The converse is false.

49. Which of the following statements about the Lebesgue integral is FALSE?

A) It is defined for a larger class of functions than the Riemann integral. B) It satisfies the Monotone Convergence Theorem. C) It satisfies the Dominated Convergence Theorem. D) It is always equal to the Riemann integral whenever both exist.

Lebesgue measure and integral on R - integrals of bounded measurable functions, comparison of Riemann and Lebesgue integrals, monotone convergence theorem, repeated integrals - Online Test

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