Lebesgue measure and integral on R - integrals of bounded measurable functions, comparison of Riemann and Lebesgue integrals, monotone convergence theorem, repeated integrals - One Line Questions

1. The Dominated Convergence Theorem (DCT) requires that the sequence of measurable functions {f_n} be dominated by an integrable function g, meaning: — |f_n(x)| ≤ g(x) for all n and x
2. 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: — ∫ f dm = lim_{n→∞} ∫ f_n dm
3. If f_n → f pointwise and |f_n| ≤ g where g is integrable, the DCT states: — ∫ f dm = lim_{n→∞} ∫ f_n dm
4. 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: — ∫ f^+ dm - ∫ f^- dm
5. If f(x, y) is non-negative and measurable on R^2, then Fubini's Theorem guarantees: — ∫∫ f(x,y) dx dy = ∫ [∫ f(x,y) dy] dx
6. If ∫∫ |f(x,y)| dx dy < ∞, then which of the following is true? — ∫∫ f(x,y) dx dy = ∫ [∫ f(x,y) dy] dx = ∫ [∫ f(x,y) dx] dy
7. 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. — 1
8. What is the Lebesgue measure of a single point in R? — 0
9. The Lebesgue measure of the set of rational numbers Q in R is: — 0
10. Let f_n(x) = 1/n for all x in [0, 1]. What is lim_{n→∞} ∫_[0,1] f_n dm? — 0
11. Let f_n(x) = x^n on [0, 1]. What is lim_{n→∞} ∫_[0,1] f_n dm? — 0
12. Let f_n(x) = 1/n * sin(nx) on [0, 1]. What is lim_{n→∞} ∫_[0,1] f_n dm? — 0
13. What is the Lebesgue integral of the Dirichlet function f(x) = 1_{Q}(x) (indicator of rationals) over [0, 1]? — 0
14. Which of the following sets is Lebesgue measurable? — The Cantor set
15. If E is a measurable set and A is any subset of E, then m(A) ≤ m(E). This property is called: — Monotonicity
16. The concept of 'bounded measurable function' is important for the initial definition of the Lebesgue integral because: — It simplifies the definition by relating it to the measure of sets.
17. The Monotone Convergence Theorem (MCT) applies to a sequence of: — Non-negative measurable functions.
18. 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? —
19. The Lebesgue integral of a constant function c over a measurable set E is: — c * m(E)
20. 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? — Functions with discontinuities on a set of positive measure
21. Which property does the Lebesgue outer measure always satisfy? — Subadditivity
22. The condition ∫ |f| dm < ∞ for a measurable function f implies that: — f is Lebesgue integrable.
23. 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: — lim ∫ f_n dm = ∫ f dm
24. A function f is Lebesgue integrable if and only if: — Both ∫ f^+ dm and ∫ f^- dm are finite.
25. The concept of 'measurable function' is fundamental to Lebesgue integration. A function f is measurable if: — Its inverse image of any measurable set is measurable.
26. Which of the following statements about the Lebesgue integral is FALSE? — It is always equal to the Riemann integral whenever both exist.
27. What is Tonelli's Theorem? — It states that for a non-negative measurable function, the repeated integrals are equal and equal to the double integral.
28. What is Fatou's Lemma? — It states that for a sequence of non-negative measurable functions {f_n}, ∫ lim inf f_n dm ≤ lim inf ∫ f_n dm.
29. What is the relationship between Riemann integrability and Lebesgue integrability for a bounded function on a closed interval? — Riemann integrability implies Lebesgue integrability.
30. The MCT is crucial for proving the integrability of functions that are defined as: — Limits of sequences of simple functions.
31. A set E in R is Lebesgue measurable if and only if for every set A in R, the following equality holds: — m*(A) = m*(A ∩ E) + m*(A ∩ E^c)
32. Which theorem provides the foundation for interchanging the order of integration in Lebesgue calculus? — Fubini's Theorem
33. The DCT is more general than the MCT because it allows for: — All of the above.
34. Which of the following is NOT a property of Lebesgue measure? — Finite subadditivity
35. What is the primary difference between repeated integrals (or iterated integrals) and multiple integrals in the context of Lebesgue integration? — Repeated integrals are computed by integrating with respect to one variable at a time, while multiple integrals are computed over a region.
36. If f is Riemann integrable on [a, b], then f is Lebesgue integrable on [a, b]. What about the converse? — The converse is true if the set of discontinuities of f has measure zero.
37. Fubini's Theorem relates multiple integrals to repeated integrals under certain conditions. What is a key condition for Fubini's Theorem to hold? — The function must be absolutely integrable.
38. Fatou's Lemma is useful for establishing the existence of integrals for sequences of functions where: — The functions are non-negative and converge pointwise.
39. What is the definition of Lebesgue outer measure of a set E in R? — The infimum of the sum of lengths of open intervals covering E.
40. Which of the following is a consequence of the Monotone Convergence Theorem? — The integral of a limit of a sequence of non-negative functions is the limit of their integrals.
41. 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: — The infimum of integrals of simple functions greater than or equal to f.
42. 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: — True
43. 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. — True
44. 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. — True
45. If f_n → f pointwise and f_n are non-negative, the MCT implies ∫ f dm ≤ lim inf ∫ f_n dm. This is: — False
46. 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. — True, for f ≥ 0.
47. If f is a bounded measurable function on R, and E is a measurable set with m(E) = 0, then ∫_E f dm is: — 0
48. 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? — Yes, the integral is 0.
49. 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]? — No, because the set of discontinuities has positive measure.
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