Lebesgue measure and integral on R - integrals of bounded measurable functions, comparison of Riemann and Lebesgue integrals, monotone convergence theorem, repeated integrals - Question Bank
1. Which of the following statements about the Lebesgue integral is FALSE?
2. If f is Riemann integrable on [a, b], then f is Lebesgue integrable on [a, b]. What about the converse?
3. 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.
4. 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:
5. If f_n → f pointwise and f_n are non-negative, the MCT implies ∫ f dm ≤ lim inf ∫ f_n dm. This is:
6. The concept of 'measurable function' is fundamental to Lebesgue integration. A function f is measurable if:
7. The Lebesgue integral of a constant function c over a measurable set E is:
8. If E is a measurable set and A is any subset of E, then m(A) ≤ m(E). This property is called:
9. Which of the following is NOT a property of Lebesgue measure?
10. The condition ∫ |f| dm < ∞ for a measurable function f implies that:
11. A function f is Lebesgue integrable if and only if:
12. 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:
13. What is the Lebesgue integral of the Dirichlet function f(x) = 1_{Q}(x) (indicator of rationals) over [0, 1]?
14. 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]?
15. 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?
16. If f is a bounded measurable function on R, and E is a measurable set with m(E) = 0, then ∫_E f dm is:
17. Which of the following is a consequence of the Monotone Convergence Theorem?
18. Fatou's Lemma is useful for establishing the existence of integrals for sequences of functions where:
19. What is Fatou's Lemma?
20. 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.
21. Let f_n(x) = 1/n * sin(nx) on [0, 1]. What is lim_{n→∞} ∫_[0,1] f_n dm?
22. Which theorem provides the foundation for interchanging the order of integration in Lebesgue calculus?
23. 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.
24. 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.
25. The concept of 'bounded measurable function' is important for the initial definition of the Lebesgue integral because:
26. 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?
27. If ∫∫ |f(x,y)| dx dy < ∞, then which of the following is true?
28. What is Tonelli's Theorem?
29. If f(x, y) is non-negative and measurable on R^2, then Fubini's Theorem guarantees:
30. Fubini's Theorem relates multiple integrals to repeated integrals under certain conditions. What is a key condition for Fubini's Theorem to hold?
31. What is the primary difference between repeated integrals (or iterated integrals) and multiple integrals in the context of Lebesgue integration?
32. The DCT is more general than the MCT because it allows for:
33. If f_n → f pointwise and |f_n| ≤ g where g is integrable, the DCT states:
34. The Dominated Convergence Theorem (DCT) requires that the sequence of measurable functions {f_n} be dominated by an integrable function g, meaning:
35. Let f_n(x) = x^n on [0, 1]. What is lim_{n→∞} ∫_[0,1] f_n dm?
36. Let f_n(x) = 1/n for all x in [0, 1]. What is lim_{n→∞} ∫_[0,1] f_n dm?
37. The MCT is crucial for proving the integrability of functions that are defined as:
38. 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:
39. The Monotone Convergence Theorem (MCT) applies to a sequence of:
40. What is the relationship between Riemann integrability and Lebesgue integrability for a bounded function on a closed interval?
41. 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?
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:
43. 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:
44. Which of the following sets is Lebesgue measurable?
45. The Lebesgue measure of the set of rational numbers Q in R is:
46. What is the Lebesgue measure of a single point in R?
47. A set E in R is Lebesgue measurable if and only if for every set A in R, the following equality holds:
48. Which property does the Lebesgue outer measure always satisfy?
49. What is the definition of Lebesgue outer measure of a set E in R?