Math 5051 Measure Theory and Functional Analysis I Homework Assignment 2

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1 Math 551 Measure Theory and Functional nalysis I Homework ssignment 2 Prof. Wickerhauser Due Friday, September 25th, 215 Please do Exercises 1, 4*, 7, 9*, 11, 12, 13, 16, 21*, 26, 28, 31, 32, 33, 36, 37. 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 Real nalysis for Graduate Students, version 2.1, by Richard F. Bass. These exercises originate from that source. 1. Suppose (X, ) is a measurable space, f : X R is a function, and {x : f(x) > r} for all r Q. Prove that f is measurable. 2. Let f : (, 1) R be such that for every x (, 1) there exist r > and a Borel measurable function g, both depending on x, such that f and g agree on (x r, x + r) (, 1). Prove that f is Borel measurable. 3. Suppose f n is a sequence of measurable functions. Prove that is a measurable set. = {x : f n(x) exists} 4. If f : R R is Lebesgue measurable, prove that there exists a Borel measurable function g such that f = g a.e. 5. Give an example of a collection of measurable non-negative functions {f α } α such that if g is defined by g(x) = sup α f α (x), then g is finite for all x but g is non-measurable. (Hint: is allowed to be uncountable.) 1

2 6. Suppose f : R R is Lebesgue measurable and g : R R is continuous. Prove that g f is Lebesgue measurable. Is this true if g is Borel measurable instead of continuous? Is this true if g is Lebesgue measurable instead of continuous? 7. Suppose f : R R is Borel measurable. Define to be the smallest σ-algebra containing the sets {x : f(x) > a} for every a R. Suppose g is measurable with respect to, namely that ( a R){x : g(x) > a}. Prove that there exists a Borel measurable function h : R R such that g = h f. 8. It is known that there exist discontinuous real-valued functions f such that ( x, y R) f(x + y) = f(x) + f(y). (1) (n example may be constructed using Zorn s lemma.) Prove that if f satisfies (1) and in addition f is Lebesgue measurable, then f is continuous. 9. Verify Equation (6.5) on textbook p.48. Namely, for measure space (X,, µ), show that if ni=1 a i χ i = m j=1 b j χ Bj for i, B j and a i, b j R, then n m a i µ( i ) = b j µ(b j ). i=1 j=1 1. Suppose f is non-negative and measurable and µ is σ-finite. Show there exist simple functions s n increasing to f at each point such that µ({x : s n (x) }) < for each n. 11. Let f be a non-negative measurable function. Prove that min(f, n) = f. 12. Let (X,, µ) be a measure space and suppose µ is σ-finite. Suppose f is integrable. Prove that given ɛ > there exists δ > such that f(x) µ(dx) < ɛ whenever µ() < δ. 13. Suppose µ(x) < and f n is a sequence of bounded real-valued measurable functions that converge to f uniformly. Prove that f n dµ f dµ. 2

3 This is sometimes called the bounded convergence theorem. (Hint: prove this with or without the dominated convergence theorem.) 14. If f n is a sequence of non-negative integrable functions such that f n (x) decreases to f(x) for every x, prove that f n dµ f dµ. 15. Let (X,, µ) be a finite measure space and suppose f is a non-negative, measurable function that is finite at each point of X, but is not necessarily integrable. Prove that there exists a continuous increasing function g : [, ) [, ) such that x g(x) = and g f is integrable. 16. State and prove a version of the Dominated Convergence Theorem for complex-valued functions. (Hint: prove this as a corollary to the dominated convergence theorem for real-valued functions.) 17. Suppose f n, g n, f, g are integrable, f n f a.e., g n g a.e., f n g n for each n, and gn g. Prove that f n f. (Hint: use the dominated convergence theorem.) 18. Give an example of a sequence of non-negative functions f n tending to pointwise such that fn but there is no integrable function g such that f n g for all n. 19. Suppose (X,, µ) is a measure space, f and each f n is integrable and non-negative, f n f a.e, and f n f. Prove that for each, f n dµ f dµ. 2. Suppose f n and f are integrable, f n f a.e., and f n f. Prove that f n f. 21. Suppose f : R R is integrable, a R, and we define Show that F is a continuous function. F (x) def = x a f(y) dy. 22. Let f n be a sequence of non-negative Lebesgue measurable functions on R. Is it necessarily true that sup If not, give a counterexample. f n dx 3 sup f n dx?

4 23. Find the it and justify your reasoning. 24. Find the it and justify your reasoning. n n ( 1 + x n) n log(2 + cos(x/n)) dx ( 1 x n) n log(2 + cos(x/n)) dx 25. Prove that the it exists and find its value: nx 2 (1 + x 2 log(2 + cos(x/n)) dx ) n 26. Prove that the it exists and determine its value: ne nx sin(1/x) dx 27. Let g : R R be integrable and let f : R R be bounded, measurable, and continuous at 1. Prove that exists and determine its value. n f (1 + x ) n n 2 g(x) dx 28. Suppose µ(x) <, f n converges to f uniformly, and each f n is integrable. Prove that f is integrable and f n f. Is the condition µ(x) < necessary? 29. Prove that k=1 1 1 (p + k) 2 = x p log x dx 1 x for p >. (Hint: use the fundamental theorem of calculus, to be proved in textbook Chapter 8, namely if f is continuous on [a, b] and F is differentiable on [a, b] with derivative f, then b a f(x) dx = F (b) F (a).) 3. Let f n be a sequence of measurable real-valued functions on [, 1] that is uniformly bounded. a. Show that if is a Borel subset of [, 1], then there exists a subsequence n j such that f n j (x) dx converges. 4

5 b. Show that if { i } is a countable collection of Borel subsets of [, 1], then there exists a subsequence n j such that i f nj (x) dx converges for each i. c. Show that there exists a subsequence n j such that f n j (x) dx converges for each Borel subset of [, 1]. 31. Let (X,, µ) be a measure space. sequence of measurable functions {f n } is uniformly integrable if, given ɛ >, there exists M such that f n (x) dµ < ɛ {x: f n(x) >M} for each n. The sequence is uniformly absolutely continuous if, given ɛ >, there exists δ > such that for each n and each with µ() < δ. f n dµ < ɛ Suppose that µ is a finite measure. Prove that {f n } is uniformly integrable if and only if sup n fn dµ < and {f n } is uniformly absolutely continuous. 32. Suppose µ is a finite measure, f n f a.e., and {f n } is uniformly integrable (see Exercise 31). Prove that f n f. (This is known as the Vitali convergence theorem.) 33. Suppose µ is a finite measure, f n f a.e., each f n is integrable, and f n f. Prove that {f n } is uniformly integrable (see Exercise 31). 34. Suppose µ is a finite measure and for some ɛ >, sup f n 1+ɛ dµ <. n Prove that {f n } is uniformly integrable (see Exercise 31). 35. Suppose {f n } is a uniformly integrable sequence of functions defined on [, 1]. Prove that there is a subsequence n j such that 1 f n j g dx converges whenever g is a real-valued bounded measurable function. 36. Suppose µ n is a sequence of measures on (X, ) such that µ n (X) = 1 for all n and µ n () converges as n for each. Call the it µ(). a. Prove that µ is a measure. b. Prove that f dµ n f dµ whenever f is bounded and measurable. 5

6 c. Prove that f dµ inf whenever f is non-neagative and measurable. f dµ n 37. Let (X,, µ) be a measure space and let f be non-negative and integrable. Define ν on by a. Prove that ν is a measure. ν() def = f dµ. b. Prove that if g is integrable with respect to ν, then fg is integrable with respect to µ and g dν = fg dµ. 38. Suppose µ and ν are finite positive measures on the Borel σ-algebra on [, 1] such that f dµ = f dν whenever f is real-valued and continuous on [, 1]. Prove that µ = ν. 39. Let B be the Borel σ-algebra on [, 1]. Let µ n be a sequence of finite measures on ([, 1], B) and let µ be another finite measure on ([, 1], B). Suppose mu n ([, 1]) µ([, 1]). Prove that the following are equivalent: a. f dµn f dµ whenever f is a continuous real-valued function on [, 1]; b. sup µ n (F ) µ(f ) for all closed F [, 1]; c. inf µ n (G) µ(g) for all open G [, 1]; d. µ n () = µ() for all B such that µ( ) =, where def = Ā o is the boundary of ; e. µ n ([, x]) = µ([, x]) for all x [, 1] such that µ({x}) =. 4. Let B be the Borel σ-algebra on [, 1]. Suppose µ n are finite measures on ([, 1], B) such that f dµn 1 f(x) dx whenever f is a real-valued continuous function on [, 1]. Suppose that g is a bounded measurable function such that the set of discontinuities of g has measure. prove that g dµ n 1 g(x) dx. 41. Let B be the Borel σ-algebra on [, 1]. Let µ n be a sequence of finite measures on ([, 1], B) such that sup n µ n ([, 1]) <. Define α n (x) = µ n ([, x]). 6

7 a. If r [, 1] is rational, prove that there exists a subsequence {n j } such that α nj (r) converges. b. Prove that there exists a subsequence {n j } such that α nj (r) converges for every rational in [, 1]. c. Let ᾱ(r) def = α n (r) for rational r [, 1]. Prove that r > s with r, s Q [, 1] implies ᾱ(r) ᾱ(s). d. Define Prove that α(x) = ᾱ(r). r x+, r Q α(x) = inf{ᾱ(r) : r > x, r Q [, 1]}. e. Let µ be the Lebesgue-Stieltjes measure associated with α. Prove that f dµ n f dµ whenever f is a continuous real-valued function on [, 1]. 7