Math 551 Measure Theory and Functional Analysis I Homework Assignment 3 Prof. Wickerhauser Due Monday, October 12th, 215 Please do Exercises 3*, 4, 5, 6, 8*, 11*, 17, 2, 21, 22, 27*. Exercises marked with (*) are especially important and you may wish to focus extra attention on those. You are encouraged to try the other problems in this list as well. Note: textbook refers to eal Analysis for Graduate Students, version 2.1, by ichard F. Bass. These exercises originate from that source. 1. Prove these basic change of variable formulas for Lebesgue integrals: If f : is integrable and a, then f(x + a) dx = f(x) dx and f(ax) dx = a 1 f(x) dx. 2. Let (X, A, µ) be a σ-finite measure space. Suppose f is non-negative and integrable. Prove that if ɛ > there exists A A such that µ(a) < and ɛ + f dµ > A f dµ. 3. Suppose A is a Borel measurable subset of [, 1], m is Lebesgue measure, and ɛ (, 1). Prove that there exists a continuous function f : [, 1] such that f 1 and m({x : f(x) χ A (x)}) < ɛ 1
4. Suppose f is a non-negative integrable function on a measure space (X, A, µ). Prove that lim tµ({x : f(x) t}) =. t 5. Let m be Lebesgue measure. Find a non-negative measurable function f on [, 1] such that but f is not integrable. lim t m({x : f(x) t}) = t 6. Suppose µ is a finite measure. Prove that a measurable non-negative function f is integrable if and only if µ({x : f(x) n}) <. n=1 7. Let µ be a measure, not necessarily σ-finite, and suppose f is real-valued and integrable with respect to µ. Prove that A = {x : f(x) } has σ-finite measure, namely that there exists F n A such that µ(f n ) < for each n. 8. ecall that a function f : is convex if whenever x, y and λ [, 1]. f(λx + (1 λ)y) λf(x) + (1 λ)f(y) a. Prove that if f is convex and x, there exists c such that f(y) f(x) + c(y x) for all y. (x, f(x)).) (So the graph of f lies above the line with slope c passing through b. Let (X, A, µ) be a measure space, suppose µ(x) = 1, and let f : be convex. Let g : X be integrable. Prove that ( ) f g dµ (This called Jensen s inequality.) 9. Suppose f : satifies ( 1 f X f g dµ. ) 1 g(x) dx f(g(x)) dx whenever g is bounded and measurable. Prove that f is convex (see Exercise 8). 2
1. Suppose g : [, 1] is bounded and measurable and 1 f(x)g(x) dx = whenever f is continuous and 1 f(x) dx =. Prove that there is some constant c such that g = c a.e. 11. Find a measurable function f : [, 1] such that (f) 1 f(x) dx and (f) 1 f(x) dx. 12. Find a function f : (, 1] that is continuous, is not Lebesgue integrable, but for which the improper iemann integral exists and is finite. lim a + (fχ (a,1]) 13. Suppose f : [, 1] that is Lebesgue integrable, f is bounded on (a, 1] for each a (, 1), and the improper iemann integral lim a + (fχ (a,1]) exists. Show that the limit is equal to (the Lebesgue integral) 1 f(x) dx. 14. Divide [a, b] into 2 n equal subintervals and pick a point x i out of each subinterval. Let µ n be the measure defined by where δ y is the point mass at y. Then b a 2 n µ n (A) = 2 n δ xi (A), f(x) µ(dx) = i=1 2 n i=1 f(x i )2 n for any bounded measurable f : [a, b], so this measure gives the iemann sum approximation to (f). a. Prove that µ n ([, x]) m([, x]) for every x [, 1]. Conclude by Exercise 7.24 on textbook p.6 (which is Exercise 39 of HW 2) that whenever f is continuous. f dµ n 3 1 f dx (1)
b. Use Exercise 7.25 on textbook p.6 (which is Exercise 39 of HW 2) to prove that if f : [a, b] is a bounded and measurable function whose set of discontinuities has measure, then the iemann sum approximation given in Equation 1 converges to the Lebesgue integral of f. This provides an alternative proof of Step 2 of Theorem 9.1 on textbook p.71. 15. Let f be a bounded, real-valued, and measurable function. Prove that if f = lim δ sup y x <δ,a y b f(y), then f = T a.e., using the notation of Theorem 9.1 on textbook p.7. Conclude that f is Lebesgue measurable. 16. Define f = lim δ inf f(y), y x <δ,a y b and let f be as in Exercise 15. a. Suppose that the set of discontinuities of a bounded real-valued measurable function f has positive Lebesgue measure. Prove that there exists ɛ > such that if A ɛ def = {x [a, b] : f(x) f(x) > ɛ}, then m(a ɛ ) >. b. Prove that U(P, f) L(P, f) > ɛm(a ɛ ) for every partition P on [a, b], using the notation of Theorem 9.1 of textbook p.7. Conclude that f is not iemann integrable. This provides an alternative proof of Step 1 of Theorem 9.1. 17. A real-valued function on a metric space is lower semicontinuous if {x : f(x) > a} is open for all a, and is upper semicontinuous if {x : f(x) < a} is open for all a. a. Prove that if f n is a sequence of real-valued continuous functions increasing to f, then f is lower semicontinuous. b. Find a bounded lower semicontinuous function f : [, 1] such that f is continuous everywhere except x = 1/2. c. Find a bounded lower semicontinuous function f : [, 1] such that the set of discontinuities of f is Q [, 1], the rationals in [, 1]. d. Find a bounded lower semicontinuous function f : [, 1] such that the set of discontinuities of f has positive measure. 4
e. Does there exist a bounded lower semicontinuous function f : [, 1] such that f is discontinuous a.e.? 18. Find a sequence f n : [, 1] [, 1] of continuous functions such that f n f but f is not iemann integrable. (This shows that there is no monotone convergence theorem or dominated convergence theorem for the iemann integral.) 19. Let M > and let B be the σ-algebra on X = [ M, M] [ M, M] 2 generated by the collection of closed rectangles {[a, b] [c, d] : M a b M, M c d M}. Suppose µ is a measure on (X, B) such that µ([a, b] [c, d]) = (b a)(d c). Prove that if f is continuous on 2 with support in X, then the Lebesgue integral of f with respect to µ is equal to the double iemann integral of f and the two iterated iemann integrals of f. 2. Let (X, A, µ) be a measure space. Say that a sequence f n of measurable real-valued functions on X is Cauchy in measure if, for all ɛ > and a >, there exists N such that m, n N µ({x : f n (x) f m (x) > a}) < ɛ. Prove that if f n is Cauchy in measure, then it converges in measure. 21. Suppose µ(x) <. Define d(f, g) def = f g 1 + f g dµ. a. Prove that d is a metric on the space of measurable functions, except that d(f, g) = only implies that f = g a.e., not necessarily everywhere. b. Prove that f n f in measure if and only d(f n, f). 22. Prove that if f n f in measure and each f n is non-negative, then f lim inf f n. n 23. Prove that if A n is measurable with µ(a n ) < for each n, and χ An f in measure, then there exists a measurable set A such that f = χ A a.e. 5
24. Suppose for each ɛ > there exists a measurable set F such that µ(f c ) < ɛ and f n converges to f uniformly on F. Prove that f n f a.e. 25. Suppose that f n and f are measurable functions such that for each ɛ >, Prove that f n f a.e. µ({x : f n (x) f(x) > ɛ}) <. n=1 26. Let f n be a sequence of measurable functions and define g n (x) def = sup f m (x) f n (x). m n Prove that if g n in measure, then f n converges a.e. 27. If (X, A, µ) is a measure space and f n is a sequence of real-valued measurable functions such that f n g dµ for every integrable g, is it necessarily true that f n in measure? Give a proof or find a counterexample. 28. Suppose (X, A, µ) is a measure space and X is a countable set. Let {f n } be a sequence of measurable functions. Prove that if f n f in measure, then f n f a.e. 6