A List of Problems in Real Analysis

Transcription

1 A List of Problems in Real Analysis W.Yessen & T.Ma December 3, 218 This document was first created by Will Yessen, who was a graduate student at UCI. Timmy Ma, who was also a graduate student at UCI, is now a post-doc at Dartmouth College, now maintains this document. Problems listed here have been collected from multiple sources. Many have appeared on qualifying exams from PhD granting universities. If you would like to add to this document, please contact Timmy. Solutions are available upon request. Certainly, it is beneficial for the reader to attempt the problems first before seeking the solutions. Should there be any problems, errors, questions, or comments please contact Timmy. 1

2 Contents 1 Measure Theory and Topology Measurable Sets (Signed) Measure Functions and Integration Measurable Functions Integrable Functions Convergence Uniform Cont./ Bounded Var./ Abs. Cont L p /L Spaces Radon-Nikodym Theorem Tonelli/Fubini Theorem

3 1 Measure Theory and Topology 1.1 Measurable Sets Problem Identify which statements are true, which are false. Give a counter-example for the false statements. 1) A finite union of open sets is an open set. 2) A countable union of open sets is an open set. 3) Any (even uncountable) union of open sets is an open set. 4) A finite union of compact sets is compact. 5) A countable union of compact sets is compact. 6) A countable union of measurable sets is a measurable set. 7) Any (even uncountable) union of measurable sets is a measurable set. 8) n 1 B i = n 1 B i 9) 1 B i = 1 B i 1) α B α = α B α Problem Let f n be a sequence of continuous functions on a complete metric space X with f n (x). Let A = {x X : inf f n (x) = }. Prove that A is a countable intersection of open sets. Problem Let O n be a countable collection of dense open subsets of a complete metric space. Prove that O n is dense. Problem Does there exist a Lebesgue measurable subset A of R such that for every interval (a, b) we have µ L (A (a, b)) = b a 2? Either construct such a set or prove it does not exist. Problem Let A be a collection of pairwise disjoint subsets of a σ-finite measure space, each of positive measure. Show that A is at most countable. Problem Let A R be a Lebesgue measurable set. Denote the length of an interval I by I. (a) Suppose that m(a I) 2 3 I for every interval I. Show that A has measure. (b) Suppose that the measure of A is positive. Show that there is an interval I such that if d < I /3, then ((I A) + d) (I A) (c) Suppose that the measure of A is positive. Show that the set A A = {x y : x, y A} contains an open interval about. Problem Suppose A j [, 1] is Lebesgue measurable with measure 1/2 for each j = 1, 2,.... Prove that there is a measurable set S [, 1] with measure 1/2 such that each x S is in A j for infinitely many j. Problem Are the irrationals an F σ set? Justify. 3

4 Problem Suppose E R is such that µ L (E) > (µ L is the Lebesgue measure on R). Prove that there exists an interval I R satisfying µ L (E I) > 3 4 µ L(I) Problem Suppose f is a function of bounded variation on [a, b]. Define G = { (x, y) R 2 : x [a, b], f(x) = y } Prove that G has Lebesgue measure. Problem Let E 1,..., E n be measurable subsets of [, 1]. Suppose almost every x in [, 1] belongs to at least k of these subsets. Prove that at least one of E 1,..., E n has measure of at least k/n. Problem Suppose that A is a subset in R 2. Define for each x R 2 ρ(x) = inf { y x : y A}. Show that B r = { x R 2 : ρ(x) r } is a closed set for each non-negative r. Is the measure of B equal to the outer measure of A? Why? Problem Let {E n } be a sequence of measurable sets in a measure space (X, Σ, µ). Suppose E n exists, i.e., inf E n = sup E n. Is it always true that µ( E n ) = µ(e n )? Does it make a difference if the measure is finite? Prove or disprove by counterexample. Problem Given a measure space (X, A, µ), let {A i } be a sequence in A. Let A = i=1 A i and suppose each x A belongs to no more than k different A i. Show that µ(a) 1 k i=1 µ(a i) (a) for k = 2 (b) for k = 1999 Problem Suppose µ L (resp. µ L ) is the Lebesgue measure (resp. outer Lebesgue measure) on R, and E R. Let E 2 = { e 2 : e E }. (a) Show that, if µ L (E) =, then µ L (E2 ) =. (b) Suppose µ L (E) <. Is it true that µ L (E2 ) <? Problem Suppose (X, A) is a measurable space, and Y is the set of all signed measures ν on A for which ν(a) < whenever A A. For ν 1, ν 2 Y, define d(ν 1, ν 2 ) = sup ν 1 (A) ν 2 (A) A A Show that d is a metric on Y and that Y equipped with d is a complete measure space. Problem (a) For any ɛ >, construct an open set U ɛ (, 1) so that U ɛ Q (, 1) and m(u ɛ ) < ɛ. (b) Let A = n=1 U c 1/n. Find m(a). Show that Ac (1/2, 5/16) Problem Show that either a σ-algebra of subsets of a set X is finite, or else it has uncountably many elements. 4

5 Problem Let r 1, r 2,... enumerate all positive rational numbers. (a) Show that R + n (r n 1 n 2, r n + 1 n 2 ) (b) Show with an example that the r n may be enumerated so that R + n (r n 1 n, r n + 1 n ) = Problem Suppose that A [, 1] is a measurable set such that m(i A) m(i)/2, for all intervals I [, 1]. Show that m(a) =. Problem Assume that E is a subset of R 2 and the distance between any two points in E is a rational number. Show that E is a countable set. (Note that this is actually true in any dimension) Problem Does there exist a nowhere dense subset of [, 1] 2 R 2 (a) of Lebesgue measure greater than 9/1? (b) of Lebesgue measure 1? Problem Let f n : R R be Borel measurable for every n N. Define E to be the set of points in R such that f n (x) exists and is finite. Show that E is a Borel measurable set. Problem Let A be a subset of R of positive Lebesgue measure. k, n N and x, y A with x y = k 2 n Prove that there exists Problem Is it possible to find uncountably many disjoint measurable subsets of R with strictly positive Lebesgue measure? Problem Let X be a non-empty complete metric space and let {f n : X R} n=1 be a sequence of continuous functions with the following property: for each x X, there exists an integer N x so that {f n (x)} n>nx is either a monotone increasing or decreasing sequence. Prove that there is a non-empty open subset U X and an integer N so that the sequence {f n (x)} n>n is monotone for all x U. Problem Give an example of a subset of R having uncountable many connected components. Can such a subset be open? Closed? Problem Let {f n : [, 1] R} n=1 be a sequence of continuous functions and suppose that for all x [, 1], f n (x) is eventually nonnegative. Show that there is an open interval I X such that for all n large enough, f n is nonnegative everywhere on I. Problem Construct an open set U [, 1] such that U is dense in [, 1], the Lebesgue measure µ(u) < 1, and that µ(u (a, b)) > for any interval (a, b) [, 1]. 5

6 Problem Is it possible for a continuous function f : [, 1] R to have (a) infinitely many strict local minima? (b) uncountably many strict local minima? Prove your answers. 1.2 (Signed) Measure Problem Let µ be a measure and let λ, λ 1, λ 2 be signed measures on the measurable space (X, A). Prove: (a) if λ µ and λ µ, then λ =. (b) If λ 1 µ and λ 2 µ, then, if we set λ = c 1 λ 1 + c 2 λ 2 with c 1, c 2 real numbers such that λ is a signed measure, we have λ µ. (c) If λ 1 µ and λ 2 µ, then, if we set λ = c 1 λ 1 + c 2 λ 2 with c 1, c 2 real numbers such that λ is a signed measure, we have λ µ. Problem Let µ be a positive Borel measure on R such that µ-a.e. for every borel set E. Show that µ =. χ E+t(x) = χ E (x) t Problem Suppose a measure m is defined on a σ-algebra A of subsets of X, and m is the corresponding outer measure. Suppose A, B X. We say that A = B if m (A B) =. Prove that = is an equivalence relation. Problem Let µ be a positive measure and ν be a finite positive measure on a measurable space (X, A). Show that ν µ then for every ɛ > there exists δ > such that for every E A with µ(e) < δ we have ν(e) < ɛ. Problem Suppose that µ and ν are two σ-finite measures on the Lebesgue σ-algebra of subsets of R d. Suppose that µ ν and ν µ. Show that there is a measurable set A such that ν(a) = and Radon-Nikodym derivative dµ dν (x) > for all x Rd \ A. Problem Let (X, σ, µ) be a measure space with µ(x) <. Given sets A i σ, i 1, prove that µ( i=1a i ) = µ( n i=1a i ). Give an example to show that this need not hold when µ(x) =. 6

7 2 Functions and Integration 2.1 Measurable Functions Problem Suppose {f n } is a sequence of measurable functions on [, 1]. For x [, 1] define h(x) = #{n : f n (x) = } (the number of indices n for which f n (x) = ). Assuming that h < everywhere, prove that the function h is measurable. Problem Let x =.n 1 n 2... be a decimal representation of x [, 1]. Let f(x) = min{n i : i N}. Prove that f(x) is measurable and a.e constant. Problem Let f n be a sequence of measurable functions on (X, µ) with f n and X f ndµ = 1 show that sup fn 1/n 1 for µ a.e x n + Problem Let f : [, 1] R be continuous function. Define the signed Borel measure µ on [, 1] by dµ = fdm Assume Prove that µ =. x n dµ =, n =, 1, 2,... [,1] 2.2 Integrable Functions Problem Let f be a nonnegative, Lebesgue measurable function on the real line, such that the function g(x) = n=1 f(x+n) is integrable on the real line. Show that f = almost everywhere. Problem Suppose f is a bounded nonnegative function on (X, µ) with µ(x) =. Show that f is integrable if, and only if, { 1 2 n µ x X : f(x) > 1 } 2 n < n= Problem Given a measure space (X, A, µ), let f be a nonnegative extended real-valued, A- measurable function on a set D A with µ(d) <. Let D n = {x D : f(x) n}. Show that f is integrable if and only if n µ(d n) < Problem Let f be a Lebesgue integrable function on the real line. Prove that f(x) sin(nx)dx = Problem Suppose (X, A, µ) is a measure space, and µ(x) <. Suppose furthermore, that f is a nonnegative measurable function on X. Prove that f is integrable if and only if the series 2 n µ({x X : f(x) > 2 n }) n= 7

8 converges. Problem Let f be a real-valued integrable function on a measure space. Let {E n } be a sequence of measurable sets such that µ(e n ) =. Show that E n fdµ =. Problem Let f be a nonnegative Lebesgue measurable function on [, 1]. bounded above by 1 and [,1] fdx = 1. Show that f = 1 a.e. on [, 1]. Suppose f is Problem Let f be real-valued continuous function on [, ) such that the improper Riemann integral f(x)dx converges. Is f Lebesgue integrable on [, )? Prove or disprove by counterexample. Problem (a) Show that an extended real valued integrable function is finite a.e. (b) If f n is a sequence of measurable functions such that f n < show that n f n(x) converges a.e. to an integrable function f and f = f n n n Problem Prove that the gamma function Γ(x) = is well defined and continuous for x >. t x 1 e t dt Problem Using the fact that Lebesgue measure m is translation invariant, that is, for every measurable set E and x R, m(e) = m(x + E), prove that for f L 1 (R), and for all t R, f(x + t)dx = f(x)dx R Problem Let f be a real-valued uniformly continuous function on [, ). Show that if f is Lebesgue integrable on [, ), then x f(x) =. Does this still hold if f is continuous, but not uniformly continuous? Prove or give a counterexample. R Problem Consider the Lebesgue measure space (R, M L, µ L ) on R. Let f be a µ L -integrable extended real-valued M L -measurable function on R. Show that f(x + h) f(x) µ L (dx) = h R Problem Let S R be closed and let f L 1 ([, 1]). Assume that for all measurable 1 E [, 1] with m(e) > we have m(e) E f S. Prove that f(x) S for a.e x [, 1] 8

9 Problem Assume f, g L 2 (R). Define A(x) = R f(x y)g(y)dy. Show that A(x) C(R) and that x + A(x) =. Problem Suppose f : R R is differentiable and f L 1 (a) Show that the Lebesgue measure of {x : f(x) > t} approaches as t. (b) Show that the additional assumption that f L 1 implies that f(x) as x. Problem Given f is integrable over [, 1]. Show that there exists a decreasing sequence of positive numbers, a n, converging to, such that a n f(a n ) =. Problem Let f be nonnegative measurable function on R. Suppose f Show that f < 1 a.e. and that 1 f L1 (R). n=1 R f n converges. 2.3 Convergence Problem Let {f n } be a sequence of continuous functions on [, 1] with f n (x) 1 for all x [, 1] and all n. Prove that: (a) If for each x [, 1] we have f n (x) =, then 1 f n (x)dx = (b) Construct a sequence of {f n } so that 1 f n(x)dx =, but for each x [, 1] the sequence {f n } diverges. Problem Let {f n } be a sequence of real-valued functions such that f n(x) 1.5 dx 1. Show that there is a subsequence f nk such that 1 x.5 f nk (x)dx converge when k. Problem Let (X, U, µ) be a measure space. Let {f n } be a sequence of nonnegative extended real-valued U-measurable functions on X. Suppose f n = f exists µ-a.e. on X and moreover f n f µ-a.e. on X. Show that fdµ = f n dµ X X Problem Let (X, A, µ) be a finite measure space. Let {f n } and f be real-valued, A-measurable functions on a set D A. Suppose that f n f in measure µ on D. Let F be a real-valued, continuous function on R. Show that F f n F f in measure µ. Problem Let (X, A, µ) be a measure space. Let {f n } and f be real-valued, A-measurable functions on a set D A. Suppose that f n f in measure µ on D. Let F be a real-valued, uniformly continuous function on R. Show that F f n F f in measure µ. 9

10 Problem Let (X, A, µ) be a measure space. Let {f n } and f be real-valued, A-measurable functions on a set D A. Suppose that f n f in measure µ on D. Let F be a real-valued, continuous function on R. And that for all n, f n, f are uniformly bounded on D. Show that F f n F f in measure µ. Problem Let f be an element and {f n } be a sequence in L p (X, A, µ), where p [1, ), such that f n f p =. Show that for every ɛ > there exists δ > such that for all n N we have f n p dµ < ɛ for every E A such that µ(e) < δ E Problem Let f be an integrable function on a measure space (X, B, µ). Show that: (a) The set {f } is of σ-finite measure. (b) If f, then f = n φ n pointwise for some increasing sequence of simple functions φ n each of which vanishes outside a set of finite measure. (c) For every ɛ > there is a simple function φ such that f φ dµ < ɛ X Problem Let {f n } and {g n } be sequences of measurable functions on the measurable set E, and let f and g be measurable functions on set E. Suppose f n f and g n g in measure. Is it true that f 3 n + g n f 3 + g in measure if (a) m(e) < (b) m(e) = Problem Let (X, A, µ) be a measure space. Let {f n } and f be extended real-valued A- measurable functions on a set D A such that f n = f on D. Then for every α R we have (1) µ {D : f > α} inf µ {D : f n α} (2) µ {D : f < α} inf µ {D : f n α} Problem Let (X, A, µ) be a measure space. Let {f n } and f be a sequence of extended real-valued A-measurable functions on a set D A with µ(d) <. Show that f n converges to in measure on D if and only if f n D 1 + f n dµ = Problem Given a measure space (X, A, µ), let {f n } and f be extended real-valued A- measurable functions on a set D A and assume that f is real-valued a.e. on D. Suppose there exists a sequence of positive numbers ɛ n such that 1. n N ɛ n < 2. D f n f p dµ < ɛ n, for some positive p <. Show that the sequence {f n } converges to f a.e. on D. Problem Suppose that g and {f n } are in L 2 [, 1], with 1 f n(x) 2 dx M < for all values of n. Suppose that f n (x) = f(x) for each x. (a) Show that f L 2 [, 1]. (b) Prove that E f n(x)dx E f(x)dx for each measurable subset E of [, 1]. 1

11 (c) Prove that 1 f n(x)g(x)dx 1 f(x)g(x)dx. Problem Let (X, A, µ) be a measure space. Let {f n } and f be real-valued A-measurable functions on a set D A. (a) Define convergence of f n to f in measure on D. (b) Suppose f n converges to two functions f and g in measure on D. Show that f = g a.e. on D. Problem Let f be a real-valued Lebesgue measurable function on [, ) such that 1. f is Lebesgue integrable on every finite subinterval of [, ). 2. x f(x) = c R 1 Show that a a [,a] fdm L = c. Problem Suppose f n (x) C 1 ([, 1]) with f n () = 1, is a sequence of functions such that sup n 1 f n(x) 2 dx 1 Show that there is a subsequence of {f n } which converges uniformly on [, 1]. (Hint: First show that {f n (x)} are equicontinuous). Hint from Will: After following the first hint above, use Ascoli s theorem. Problem If f n (x) and f n (x) f(x) in measure then f(x)dx inf f n (x)dx. (Remark: this is not just Fatou s Lemma). Problem Let (X, A, µ) be a finite measure space. Let {f n } be an arbitrary sequence of real-valued A-measurable functions on X. Show that for every ɛ > there exists E A with µ(e) < ɛ and a sequence of positive real numbers {a n } such that a nf n (x) = for x X \ E Problem Let (X, A, µ) be a finite measure space. Let f and {f n } be real-valued A-measurable functions on X and f, f n L 2 (X, A, µ). Suppose 1. f n = f, µ-a.e. 2. f n 2 C for all n N. Show that f n f 1 = Problem Suppose m(x) < and f n converges to f in measure on X and g n converges to g in measure on X. Prove that f n g n converges to fg in measure on X. (Note from Timmy: Give a counter-example to show that this fails if m(x) = ) Problem Suppose that f n (x) is a sequence of measurable functions such that f n (x) converge to f(x) a.e. If for each ɛ >, there is C < such that f n(x) >C f n(x) dx < ɛ. Show that f(x) is integrable on [, 2]. Note from Will: That s exactly how this problem appeared on the Winter 2 qual. Note that C does not depend on n. To see this, find an example where C depends on n and on ɛ, such that the statement of the problem fails to hold. 11

12 Problem Show that the Lebesgue Dominated Convergence Theorem holds if almost everywhere convergence is replaced by convergence in measure. (Hint from Will: First show that a.e. convergence can be replaced by convergence in measure in Fatou s lemma). Problem Let (X, Σ, µ) be a finite measure space, and let F denote the set of measurable functions,where we identify functions that are equal µ-a.e. Define f g δ(f, g) = 1 + f g dµ X for f, g F. Prove that δ is a metric on the set F, and that (F, δ) is a complete metric space. Prove also that convergence is equivalent to convergence in this metric space. Problem Suppose that {f n } is a sequence of real-valued Lebesgue measurable functions such that 1 f n (x) x dx when n. Is it true that f n (x) converges to x for almost every x in (, 1)? Why? Problem Let (X, M, µ) be a measure space. (a) Suppose µ(x) <. If f and f n are M-measurable functions with f n f a.e., prove that there exist sets H, E k M such that X = H ( k=1 E k), µ(h) =, and f n f uniformly on each E k. (b) Is the result of (a) true if (X, M, µ) is σ-finite? Problem Consider the real valued function f(x, t), where x E R n and t I where I is an interval in the real line. Suppose 1. f(x, ) is integrable over I for all x E. 2. There exists an integrable function g(t) on I such that f(x, t) g(t) for all x E and t I. 3. For some x E the function f(, t) is continuous at x for all t I. Then show that the function F (x) = I f(x, t)dt is continuous at x. Problem (a) Let {c n,i : n N, i N} be an array of nonnegative extended real numbers. Show that inf c n,i inf c n,i i N i N (b) Show that if {c n,i } is an increasing sequence for each i N, then c n,i = i N i N c n,i Problem Let f n : [, ) [, ) be Lebesgue measurable functions such that f 1 f 2. Suppose f n (x) f(x) as n, for a.e. x [, 1]. (a) Assume in addition that f 1 L 1 1 ([, 1]). Prove that f ndx = 1 fdx. (b) Show that this is not necessarily true if f 1 / L 1 ([, 1]). 12

13 Problem Let {q k } be all the rational numbers in [, 1]. Show that 1 k 2 converges a.e. in[, 1] x q k k=1 Problem Evaluate the following a) π b) e x2 +3x n 2 x 2 cos 3x n dx 2x dx Problem Find a sequence of pointwise convergent measurable functions f n : [, 1] R such that f n does not converge uniformly on any set X with m(x) = 1. Problem Does there exist a sequence of functions f n C([, 1]) such that for all x [, 1] and f n(x) = n + max f n(x)? n + x [,1] Either construct an example or prove such a sequence does not exist. Problem Suppose that f L 1 (R). Show that for any α > n + n α f(nx) =, for a.e x Problem Suppose that {a n } is a sequence of nonnegative numbers and {E n } a sequence of measurable subsets of [, 1]. Assume that there exists δ > such that m(e n ) δ and Show that g(x) = a n χ En < for a.e x [, 1]. n=1 a n < Problem Let f be a nonnegative measurable function on R. Prove that if is integrable, then f = a.e. n=1 n= f(x + n) Problem For each n 2, define f n : [, 1] R as follows: { n 2 for x [ i f n (x) = n, i n + 1 ] and i =, 1,..., n 1 n 3. otherwise (a) Show that n + f n = for a.e x [, 1] (b) For g C[, 1], what is n + [,1] f n(x)g(x)dx? (c) Is your answer to part (b) still valid for all g L (, 1)? Justify 13

14 Problem Suppose that f n L 4 (X, µ) with f n 4 1 and f n µ-a.e on X. Show that for every g L 4/3 (X, µ), f n (x)g(x)dµ(x) as n X Problem Either prove or give a counterexample: If a sequence of functions f n on a measure space (X, µ) satisfies X f n dµ 1 n 2, then f n µ-a.e. Problem For a nonnegative function f L 1 ([, 1]), prove that Problem Compute Justify each step. 1 n f(x)dx = m({x : f(x) > }) n sin(x/n) x(1 + x 2 ) dx. Problem Give an example of a sequence of functions, f n in L 1 [, 1] such that f n 1 = but f n does not converge to almost everywhere. Problem Show that if a sequence, f k in L 1 [, 1] satisfies f k 1 2 k for k 1, then f k a.e. Problem Let f C[, 1] be real-valued. Prove that there is a monotone increasing sequence of polynomials {p n (x)} n=1 converging uniformly on [, 1] to f(x). Problem Suppose f : [, 1] R continuous (a) Prove that 1 xn f(x)dx = (b) Prove that n 1 xn f(x)dx = f(1) Problem Suppose g(x) = g n (x) for x [, 1] where {g n } are positive and continuous on [, 1]. Also suppose that 1 g ndx = 1. (a) Is it always true that 1 (b) Is it always true that 1 g(x)dx 1? g(x)dx 1? Problem Let f(x, y) be defined on the unit square S = {(x, y) R 2 : x 1 and y 1} and suppose that f has the following properties: (i) For each fixed x, the function f(x, y) is an integrable function of y on the unit interval. (ii) The partial derivative f x exists at every point (x, y) S, and is a bounded function on S. Show that: (a) The partial derivative f is a measurable function of y for each x. (b) x d dx 1 f(x, y)dy = 14 1 f (x, y)dy x

15 Problem Fatou s Lemma can be written as: Let f n be a sequence of nonnegative measurable functions. Then inf f n dx inf f n dx. Show that this version is equivalent to: If f n is a sequence of nonnegative measurable functions, and f n = f a.e, then fdx inf f n dx. Problem Let {f n } be a sequence of nonnegative measurable functions on X that converges pointwise on X to f, and f integrable over X. Show that if X f = X f n, then E f = E f n, for any measurable E X. Problem (a) Suppose f n is a sequence of nonnegative measurable functions on E and that it converges pointwise to f on E, and that f n f on E for all n. Show that E f n E f. (b) Let f be a nonnegative measurable function on R. Show that n n f = 1/n n Problem Evaluate 1 + n 2 x 2 + n 6 x 8 dx Problem Suppose f and g are real-valued on R, such that f is µ L -integrable, and g belongs to C (R) (compact support). For c > define g c (t) = g(ct). Prove that a) c R fg cdµ L = b) c R fg cdµ L = g() R fdµ L R f Problem Prove or give a counterexample: Suppose X is a finite measure space. Let {f n } be a sequence of measurable functions on X that converges in measure on X to real-valued function f. Then for each ɛ >, there is a closed set F contained in X for which f n f uniformly on F and m(x \ F ) < ɛ. Problem Let E be of finite measure. Suppose the sequence of functions {f n } is uniformly integrable over E. If f n converges to f in measure on E, then f is integrable over E and E f n = E f. Problem Let µ be the Lebesgue measure on R. Let f n = 1 n χ [,n]. Prove that f n f = a.e., but inf f n dµ < fdµ Does this contradict Fatou s Lemma? R R 15

16 2.4 Uniform Cont./ Bounded Var./ Abs. Cont. Problem a) Let f be absolutely continuous on [ɛ, 1] for each ɛ >. Let f be continuous at and of bounded variation on [, 1]. Prove that f is absolutely continuous on [, 1]. (b) Give an example of continuous function of unbounded variation on [2, 3]. Problem Let (R, U, µ) be the Lebesgue measure space, and consider an extended real-valued (U)-measurable function f on R. Let B r (x) = {y R : y x < r}, with r > fixed. Define the function g on R via g(x) = f(y)dµ(y) B r(x) for x R. (a) Suppose f is locally µ-integrable on R. Show that g is a real-valued, continuous function on R. (b) Show that if f is µ-integrable on R, then g is uniformly continuous on R. Problem Suppose that f L 1 [, 1]. Let F (x) = x f(t)dt. Let φ be a Lipschitz function. Show that there exists g L 1 [, 1] such that φ(f (x)) = x g(t)dt. Problem Let g be an absolutely continuous monotone function on [, 1]. Prove that, if E [, 1] is a set of Lebesgue measure zero, then the set g(e) is also of Lebesgue measure zero. Problem Let {f n } be a sequence of real-valued functions and f be a real-valued function on [a, b] such that f n (x) = f(x) for x [a, b]. Show that Va b (f) inf V b a (f n ) Problem Let f be a real-valued, continuous function of bounded variation on [a, b]. Suppose f is absolutely continuous on [a + η, b] for every η (, b a). Show that f is absolutely continuous on [a, b]. Problem Let f be a real-valued function on (a, b) such that the derivative f exists and f (x) M for x (a, b) for some M. Let µ L be the Lebesgue outer measure on R. Show that for every subset E of (a, b) we have µ L(f(E)) Mµ L(E) Problem Suppose µ is the Lebesgue measure on R, E is a Lebesgue measurable subset of R, and let { } µ(e [x, x + ɛ]) F = x E : = 1 ɛ ɛ Show that µ(e \ F ) = Problem Suppose f L 1 (, ), and let g(x) = x+1 x 1 16 f(t)dt

17 (a) Prove that g(t) dt 2 f(t) dt (b) Prove that g is absolutely continuous (that is, its restriction on any closed subinterval of R is absolutely continuous). Problem Prove that the discontinuity points of a real-valued function f are at most countable if (a) f is a decreasing function on [, 1]. (b) f is a function of bounded variation on [, 1]. Problem Let C([, 1], ρ) be a space of continuous functions on [, 1] with a uniform metric ρ. Let F : C([, 1]) C([, 1]) be given by Prove that F is uniformly continuous. (a) (F f)(t) = (b) (F f)(t) = t n=1 e tu sin(f(u))du 1 n! 1 f ( ) tu du n Problem (a) Find V ar 5(cos(x)). (b) Give an example of a function f such that V ar2 4 (f) =. (c) Give an example of a function f such that V ar2 4 (f) = 2. Problem Let f be absolutely continuous on [, 1] and P (x) a polynomial. Show that P (f(x)) is absolutely continuous on [, 1]. Problem Suppose that f is continuous and of bounded variation on [, 1]. Prove that n ( ) i f f n i=1 ( ) i 1 2 = n Problem Suppose f is Lipschitz continuous in [, 1], that is f(x) f(y) L x y for some constant L. Show that (a) m(f(e)) = if m(e) = (b) If E is measurable, then f(e) is also measurable. Problem Assume that f is absolutely continuous, and that f and f are both in L 1 (R). Let g(x) = f(x + k). k= Show that g L (I) for any bounded interval I. (Hint: first prove for intervals with length 1/2) 17

18 2.5 L p /L Spaces Problem Let X be a measure space and let µ be a positive Borel measure over X, with µ(x) <. Let f L (X, µ). For each positive integer n, we let α n = f(x) n dµ(x) (a) Show that α n+1 α n = f if f >. (b) Use part (a) to prove that (α n ) 1/n = f. X Problem Let (X, U, µ) be a measure space. Let f L p (X, U, µ) L q (X, U, µ) with 1 p < q <. Show that f L r (X, U, µ) for all p r q. Problem Let f L p ([, 1]), p 1. (a) Prove that t 1 +(t 1) 1 p 1 t 1 f(s)ds = (b) Suppose x 2 f 5 dx <. Prove that t t 6/5 t f(x)dx =. Problem Suppose that 1 x 1 f 3 dx <. Show that t t 1 t f(x)dx = Problem Let 1 p < q <. Which of the following statements are true and which are false? Justify. 1. L p (R, M L, µ L ) L q (R, M L, µ L ) 2. L q (R, M L, µ L ) L p (R, M L, µ L ) 3. L p ([2, 5], M L, µ L ) L q ([2, 5], M L, µ L ) 4. L q ([2, 5], M L, µ L ) L p ([2, 5], M L, µ L ) Problem Let f L 3 2 ([, 5], M L, µ L ). Prove that 1 t t 1 3 t f(s)ds = Problem Let p, q [1, ). (a) Show that L p (, ) is not a subset of L q (, ) when p q. (b) If f L p (R) and f is uniformly continuous on R, show that x f(x) =. Is the result still true if f is only assumed to be continuous? Problem Let g(x) be measurable and suppose b a f(x)g(x)dx is finite for any f(x) L2. Prove that g(x) L 2. (Note from Timmy: This is kind of an unfair question in that you have to use the uniform boundedness property from Functional Analysis, what they probably forgot to say was that instead of finite, that the integral of fg is uniformly bounded.) Problem If f(x) L p L for some p <, show that (a) f(x) L q for all q > p. (b) f = q f L q 18

19 Problem If f n (x), f(x) L 2, and f n (x) f(x) a.e., then f n f L 2 if and only if f n L 2 f L 2. Remark: You can actually prove the more general case, replacing 2 with p. Problem Let f be a Lebesgue measurable function on [, 1] and let < f(x) < for x [, 1]. Show that { } { } 1 1. f Justify each step. fdµ [,1] [,1] Problem Consider L p ([, 1]). Prove that f p is increasing in p for any bounded measurable function f. Prove that f p f when p. Problem Suppose 1 p, f L 1 (R, dx) and g L p (R, dx). Define the function f g by (f g)(x) = f(x y)g(y)dy, whenever it makes sense. Show that f g is defined almost everywhere, and that f g L p (R, dx) with f g p f 1 g p Problem Let f n f in L p, with 1 p <, and let g n be a sequence of measurable functions such that g n M < for all n and g n g a.e.. Prove that g n f n gf in L p. Problem Let E be a measurable subset of the real line. Prove that L (E) is complete. Problem Let (X, A, µ) be a measure space and let f be an extended real-valued A-measurable function on X such that X f p dµ < for some p (, ). Show that λ λp µ {x X : f(x) λ} = Problem Show that L 3 ([ 1, )) L 2 ([ 1, )) L 5 ([ 1, )) Problem Suppose that f L p ([, 1]) for some p > 2. Prove that g(x) = f(x 2 ) L 1 ([, 1]). Problem Suppose f is a Borel measurable function and 1 x 1/2 f(x) 3 dx <. Show that t t t 5/6 f(x)dx = Problem If (X, µ) is a finite measure space, 1 < p 1 < p 2 <, f, and prove that f L p 1 sup λ p 2 µ(x : f(x) > λ) < λ> Problem Let < r < p < q. For f in L p (R), show that f = g + h where g L r (R) and h L q (R). Furthermore, show that given b >, prove that g, h can be chosen so that g r r b r p f p p and h q q b q p f p p. 19

20 Problem Let A be the collection of functions f L 1 (X, µ) such that f 1 = 1 and X fdµ =. Prove that for every g L (X, µ), fgdµ = 1 (ess sup g ess inf g) 2 sup f A X Problem Let f(x) be continuous function on [, 1] with a continuous derivative f (x). Given ɛ >, prove that there is a polynomial p(x) so that f(x) p(x) + f (x) p (x) < ɛ Problem Let X be a σ-finite measure space, and f n : X R a sequence of measurable function on it. Suppose f n in L 2 and L 4. Give a proof or counterexample to the following: (a) f n in L 1 (b) f n in L 3 (c) f n in L 5 Problem Let f be a nonnegative measurable function and ess sup f = M >. Then, if µ(x) <, prove f n+1 f n = M Problem Let k + m = km and let f and g be nonnegative measurable functions. Show that if < k < 1 or k < then fg ( f k dµ) 1/k ( g m dµ) 1/m Problem Let < p < 1 and f, g, f, g L p. Show that f + g p f p + g p Problem Suppose 1 p, f L p ([, 1]), and h(t) is the Lebesgue measure of the set {x [, 1] : f(x) > t} for t <. Show that h(t)dt < if 1 < p. Is this still true for p = 1? Prove or find a counterexample. 2.6 Radon-Nikodym Theorem Problem Let ν be a finite Borel measure on the real line, and set F (x) = ν {(, x]}. Prove that ν is absolutely continuous with respect to the Lebesgue measure µ if, and only if, F is an absolutely continuous function. In this case show that its Radon-Nikodym derivative is the derivative of F, that is: dν dµ = F almost everywhere. Problem Let (X, Σ, µ) be a σ-finite measure space, and let Σ be a sub-σ-algebra of Σ. Given an integrable function f on (X, Σ, µ), show that there is a Σ -measurable function f, such that fgdµ = f gdµ X 2 X

21 for every Σ -measurable function g such that fg is integrable. (Hint: Use the Radon-Nikodym Theorem). Problem Let f, g be extended real-valued, measurable functions over a σ-finite measure space (X, A, µ). Show that if E fdµ = E gdµ for all E A, then f = g on X µ-a.e. Problem Let M denote the Lebesgue measurable sets on the real line. Consider two measures on M: Lebesgue measure m and the counting measure τ. Show that m is absolutely continuous with respect to τ, but that dm dτ does not exist. Does this contradict the Radon-Nikodym theorem? Problem Prove the following statement. Suppose that F is a sub-σ-algebra of the Borel σ-algebra on the real line. If f(x) and g(x) are F -measurable and if fdx = gdx for all A F A A then f(x) = g(x) a.e. Note from Will: This is precisely how the above problem appeared on the Winter 2 real analysis qual. Note that the statement is in general false, as it appears above. One needs to add the assumption that F is σ-finite. So prove the following: (a) σ-finiteness of F does NOT follow from it being a sub-σ-algebra of a σ-finite σ-algebra. (b) Show that if σ-finiteness is not required, then the statement of the problem is in general false. (c) Prove the statement of the problem with the additional assumption that F is σ-finite. Problem Let µ and ν be two measures on the same measurable space, such that µ is σ-finite and ν is absolutely continuous with respect to µ. (a) If f is a nonnegative measurable function, show that fdν = f [ ] dν dµ dµ (b) If f is a measurable function, prove that f is integrable with respect to ν if and only if f is integrable with respect to µ, and that in this case we also have equality as above. [ ] dν dµ Problem Let λ µ and µ λ be σ-finite measures on (X, X ). What is the relationship between the Radon-Nikodym derivatives dµ dλ dλ and dµ? Problem Assume that ν and µ are two finite measures on a measurable space (X, M). Prove that ν µ (ν nµ)+ = Problem Suppose that we have ν 1 µ 1 and ν 2 µ 2 for positive measures ν i and µ i on measurable spaces (X i, M i ), (i = 1, 2). Show that we have ν 1 ν 2 µ 1 µ 2 and d(ν 1 ν 2 ) d(µ 1 µ 2 ) (x, y) = dν 1 dµ 1 (x) dν 2 dµ 2 (y) 21

22 Problem Suppose µ and ν are σ-finite measures on a measurable space (X, A), such that ν µ and ν µ ν. Prove that µ({x X : dν dµ = 1}) = 2.7 Tonelli/Fubini Theorem Problem Let f(x) be a locally integrable function on (, ). We define F (x) = 1 x x f(t)dt for x >. Prove that if f(x) > on (, ) and f(x)dx <, then F (x)dx =. Problem Let (X, U, µ) and (X, V, ν) be the measure spaces given by X = Y = [, 1] U = V = B [,1] where B [,1] is the Borel σ-algebra, and µ is the Lebesgue measure, ν is the counting measure. Consider the product measurable space (X Y, σ(u V)) and a subset in it defined by E = {(x, y) X Y : x = y}. (a) Show that E σ(u V). (b) Show that [ X Y ] [ ] χ E dν dµ χ E dµ dν Y X (c) Why is Tonelli s Theorem not applicable? Problem Let f n (x) = n 2 χ [ 1 n, 1 ]. Prove that for g L1 (R), n f n (y x)g(x)dx g(y) dy as n Problem Let f be a real valued measurable function on the finite measure space (X, Σ, µ). Prove that the function F (x, y) = f(x) 5f(y) + 4 is measurable in the product measure space (X X, σ(σ Σ), µ µ), and that F is integrable if and only if f is integrable. Problem Let f L 1 (R, M L, µ L ). With h > fixed, define a function φ h on R by setting φ h (x) = 1 f(t)dµ L (t) for x R 2h [x h,x+h] (a) Show that φ h is M L -measurable on R. (b) Show that φ h is continuous on R (hint from Will: actually continuity will imply Borel measurability, hence measurability). (c) Show that φ h L 1 (R, M L, µ L ) and φ h 1 f 1 22

23 Problem Let (X, M, µ) be a complete measure space and let f be a nonnegative integrable function on X. Let b(t) = µ {x X : f(x) t}. Show that fdµ = b(t)dt Problem If g is a Lebesgue measurable real function on [, 1] such that the function f(x, y) = 2g(x) 3g(y) is Lebesgue integrable over the square [, 1] 2, show that g is Lebesgue integrable over [, 1]. Problem Let A = {(x, y) [, 1] 2 : x + y Q, xy Q}. Find A y 1/2 sin xdm 2. Problem Let f be Lebesgue integrable on (, 1) For < x < 1 define g(x) = 1 Prove that g is Lebesgue integrable on (, 1) and that 1 x g(x)dx = t 1 f(t)dt. 1 f(x)dx Problem Evaluate x exp( x 2 (1 + y 2 ))dxdy Justify each step. 23