Analysis Test Bank

July 24, 2009

This document is the Syllabus for the Common Core Analysis Comprehensive Exam. It is followed by the Test Bank of sample problems for review. In this document, functions are understood to be defined on an abstract measure space, or else on either the real line or real n–dimensional Euclidean space. Functions are real valued unless stated to the contrary.

Prerequisites

Graduate students are presumed to have preparation in advanced calculus, such as Math 4031, 4032, and 4035 at LSU. Not every student will have learned each of the topics below, so it is a good idea to round out your understanding as needed by means of reading or auditing one or more of these advanced calculus courses.

• From Math 4031: Continuous functions, uniform continuity, pointwise versus uniform convergence, compact sets, and the Riemann integral.

• From Math 4032: Differentiable functions, infinite series of constants and functions, Taylor’s theorem, C^{∞}–functions, and the Weierstrass approximation theorem.

• From Math 4035: The topology of Euclidean space, derivatives as linear transformations, Jacobians, the inverse– and implicit–function theorems. The topics in this list are particularly helpful as preparation for Math 7550—differential geometry.

Major Test Topics

(a) Measure spaces: Fields, Borel Fields and Measures

• Monotone Classes and Borel Fields

• Additive Measures, Outer Measures

• E. Hopfs Extension Theorem

(b) Lebesgue Measure

• Measurable Sets, Borel Sets, and the Real Line

• Measure Spaces and Completions, Semimetric Space of Measurable Sets

• Lebesgue Measure in R^{n}; comparison with Jordan Measure in R^{n}

(c) Measurable Functions (on measure spaces)

• Measurable Functions, Limits of Measurable Functions

• Simple Functions and Egoroffs Theorem; Convergence in Measure;

Lusin’s Theorem

(d) The Integral (on measure spaces)

• Simple Functions; the Class of nonnegative Measurable Functions, and the Class L of Lebesgue Integrable Functions

• Lebesgue Dominated Convergence Theorem; the Monotone Convergence theorem, and Fatous lemma

• L^{1}(X, 𝔄, μ) as a Complete Normed Linear Space; Complex Valued

Functions

(e) Product Measures and Fubinis Theorem (for products of measure spaces)

• Product Measures, Fubinis Theorem, Comparison of Lebesgue and Riemann Integrals

• Applications of Fubini’s theorem to L^{1} convolution, Fourier trans-

form, etc.

(f) Functions of a Real Variable

• Functions of Bounded Variation; Fundamental Theorem for the Lebesgue Integral; Lebesgue’s Theorem and Vitali’s Covering Theorem. The Cantor function.

• Absolutely Continuous & Singular Functions

• Convex functions

(g) General Countably Additive Set Functions • Hahn Decomposition Theorem

• Radon–Nikodym Theorem

• Lebesgue Decomposition Theorem

Suggested Reading

For Graduate Real Analysis:

• D.S. Bridges, Foundations of Real and Abstract Analysis, Springer, 1997 • P. Halmos, Measure Theory, D. Van Nostrand Company, Inc., 1950.

• A. Browder: Mathematical Analysis, Springer, 1996.

• I.P. Natanson, Theory of Functions of a Real Variable, Vol. 1, 1955.

• L. Richardson, Measure and Integration: a Concise Introduction to Real Analysis, John Wiley & Sons, 2009.

• H.L. Royden, Real Analysis, Macmillan, 1988.

• W. Rudin, Real and Complex Analysis, McGraw-Hill.

• G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 1963.

• E.M. Stein and R. Shakarchi, Real Analysis-Measure Theory, Integration, & Hilbert Spaces, Princeton U. Press, 2005.

• R.L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis. Marcel Dekker, 1977.

For Advanced Calculus/Undergraduate Analysis Review:

• L. Richardson, Advanced Calculus: An Introduction to Linear Analysis, John Wiley & Sons, 2008.

• Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1964.

Analysis Test Bank

Problems representing the core topics of math 7311 (Analysis I) are found in sections II, III, and IV, below.

1 Preliminaries: Advanced Calculus

These topics are found in Math 4031, 4032, and 4035. Unless otherwise stated, functions are real valued and defined on the real line and/or real n-space. In this section only, the integral is meant exclusively in the sense of Riemann. Students who have not studied these topics in undergraduate analysis may need to learn them independently as graduate students. It is possible also to audit Math 4031, 4032, and 4035.

I.1 (a) Let f_{n} : ℝ → ℝ be given by f_{n}(x) = \frac{x}{n} (n ∈ ℕ). Show that the sequence (f_{n})_{n∈ℕ} is pointwise convergent on ℝ, uniformly convergent on [0, 1], but not uniformly convergent on ℝ.

(b) Let f_{n} : [0, 1] → ℝ be given by f_{n}(x) = \frac{1}{1+nx^{2}} (n ∈ ℕ). Show that (f_{n})_{n∈ℕ} is a bounded subset of C[0, 1] and that no subsequence of (f_{n})_{n∈ℕ} converges in C[0, 1].

I.2  Identify all subsets of [0, 1] on which ∑_{n=0}^{∞} x^{n} converges uniformly. Explain.

I.3 Recall that a step function is finite linear combination of characteristic functions of intervals.

(a) Show that every continuous function on [0, 1] is a uniform limit of step functions.

(b) Is the converse true?

I.4 Prove or disprove:

(a) The product of two uniformly continuous functions on ℝ is also uniformly continuous.

(b) The product of two uniformly continuous functions on [0, 1] is also uni-

formly continuous.

I.5 Prove or disprove the following two statements:

(a) If ∑_{n=1}^{∞} a_{n} converges, then ∑_{n=1}^{∞} a_{n} cos nx converges pointwise everywhere on ℝ.

(b) If ∑_{n=1}^{∞} a_{n} converges absolutely, then ∑_{n=1}^{∞} a_{n} cos nx converges to a con-

tinuous function on ℝ.

I.6 Let C_{0,0}[0, 1] be the space of all continuous real functions f on the interval [0, 1] satisfying f(0) = f(1) = 0. Let P_{0,0} be the subspace of polynomials in C_{0,0}[0, 1]. Show that P_{0,0} is dense in C_{0,0}[0, 1] in the sup norm.

I.7 Let (x_{n})_{n∈ℕ} be a sequence of real numbers. Prove that the following are equivalent.

(a) lim_{n→∞} x_{n} = a.

(b) Every subsequence of (x_{n})_{n∈ℕ} contains a subsequence that converges to

a.

I.8  Prove: If f ∈ C[0, 1] and ∫_{0}^{1} f(x)e^{−nx} dx = 0 for all n ∈ ℕ_{0}, then f = 0.

I.9 Let f_{n} : [1, ∞) → ℝ be defined by f_{n}(x) := \frac{n+1}{n}e^{−nx} (n ∈ ℕ). Show that the series ∑_{k=1}^{∞} f_{k} converges uniformly to a continuous function.

I.10 f_{n} : ℝ → ℝ be defined by f_{n}(x) := (sin x)^{n} (n ∈ ℕ). Does (f_{n})_{n∈ℕ} converge

uniformly? I.11 Let f_{n} : [0, ∞) → ℝ be defined by f_{n}(x) := (x/n)e^{−(x/n)} (n ∈ ℕ).

(a) Determine f(x) = lim_{n→∞} f_{n}(x). Show that the sequence (f_{n})_{n∈ℕ} converges uniformly to f on [0, a] for any nonnegative real number a. Does the sequence converge uniformly to f on [0, ∞)? Justify your answer.

(b) Show that

lim_{n→∞} ∫_{0}^{a} f_{n}(x) dx = ∫_{0}^{a} f(x) dx,

but that

lim_{n→∞} ∫_{0}^{∞} f_{n}(x) dx ≠ ∫_{0}^{∞} f(x) dx.

I.12 Let f_{n}(x) = ∑_{i=0}^{n−1} \frac{1}{n}f (x + \frac{i}{n}), where f is a continuous function on IR. Show that

the sequence of functions (f_{n})_{n∈ℕ} converges uniformly on every finite segment [a, b] to the function F (x) = ∫_{x}^{x+1} f(s) ds.

I.13 Does e^{z}(x^{2} + y^{2} + z^{2}) − \sqrt{1 + z^{2}} + y = 0 have a solution z = f(x, y), where f

is continuous at (x = 1, y = 0) and f(1, 0) = 0? Explain carefully. I.14 Let f : ℝ → ℝ be an infinitely differentiable function.

(a) Use Taylor’s formula with remainder to show that, given x and h, f′(x) = (f(x + 2h) − f(x))/2h − hf′′(ξ) for some ξ.

(b) Assume f(x) → 0 as x → ∞, and that f′′ is bounded. Show that f′(x) → 0

as x → ∞.

I.15 Prove or give a counterexample: If f is a uniform limit of polynomials on [−1, 1], then the Maclaurin series of f converges to f in some neighborhood of 0.

I.16 Let f(x) = x^{k} sin(1/x) if x ≠ 0 and f(0) = 0.

(a) If k = 2, show that f is differentiable everywhere but f′ fails to be continuous at some point.

(b) If k = 3, does f have a second derivative at all points? If so, is f′′ a

continuous function? Give your reasons.

I.17 Let f be defined on ℝ^{3} by f(x, y, z) = x^{2}+4y^{2}−2yz−z^{2}. Show that f(2, 1, −4) = 0 and f_{z}(2, 1, −4) ≠ 0, and that there exists therefore a differentiable function g in a neighborhood of (2, 1) in ℝ^{2}, such that f(x, y, g(x, y)) = 0. Find g_{x}(2, 1) and g_{y}(2, 1). What is the value of g(2, 1)?

I.18 Suppose that a function f is defined on (0, 1] and has a finite derivative with |f′(x)| < 1. Define a_{n} := f(1/n) for n = 1, 2, 3, . . .. Show that lim_{n→∞} a_{n} exists.

I.19 Define a function f on ℝ by

f(x) = \{ \begin{matrix}e^{−1/x^{2}}, if x > 0 \\ 0 , if x ≤ 0 \end{matrix}

(a) Check whether f is infinitely differentiable at 0, and, if so, find f^{(n)}(0), n = 1, 2, 3, · · ·. Show details.

(b) Does f have a power series expansion at 0?

(c) Let g(x) = f(x)f(1−x). Show that g is a nontrivial infinitely differentiable

function on ℝ which vanishes outside (0, 1).

I.20 Prove that a function f : ℝ^{n} → ℝ is continuous at x = (x_{1}, ⋯, x_{n}) ∈ ℝ^{n} if the

partial derivatives f_{x_{1}}, …, f_{x_{n}} exist and are bounded in a neighborhood of x. I.21 Let f_{N}(x) = ∑_{n=1}^{N} a_{n} sin(nx) for a_{n}, x ∈ ℝ. If ∑_{n=1}^{∞} na_{n} converges absolutely,

show that (f_{N})_{N∈ℕ} converges uniformly to a function f on ℝ, and that (f′_{N})_{N∈ℕ} converges uniformly to f′ on ℝ.

I.22 Let f be a twice continuously differentiable real-valued function on ℝ^{n}. A point x ∈ ℝ^{n} is a critical point of f if all partial derivatives of f vanish at x (i.e., ▽f(x) = 0), a critical point x is nondegenerate if the n × n matrix

[\frac{∂^{2}f}{∂x_{i}∂x_{j}} (x)]

is nonsingular. Let x be a nondegenerate critical point of f. Prove that there is an open neighborhood of x which contains no other critical points. (i.e., the nondegenerate critical points are isolated.)

I.23 Show that a function f(x) = e^{−x} + 2e^{−2x} + … + ne^{−nx} + … is continuous on (0, ∞).

I.24 Let f : ℝ → ℝ be a twice differentiable function with bounded first and second derivatives. By considering the Taylor expansion of f, show that: ∥f′∥_{∞} ≤ \frac{1}{h}∥f∥_{∞} + h∥f′′∥_{∞} for every h > 0. By minimizing over h, show that ∥f′∥_{∞} ≤ 2\sqrt{∥f∥_{∞}∥f′′∥_{∞}}, where ∥g∥_{∞} denotes sup_{x∈ℝ} |g(x)|.

I.25 Let f be a real-valued differentiable function on an interval (a, b). Show that f

is Lipschitz continuous if and only if f has bounded derivative. I.26 Let E be a nonempty subset of ℝ. Let c ∈ ℝ. Show sup E = c if and only if

x ≤ c for all x ∈ E and for each ϵ > 0, there is an x ∈ E with x + ϵ > c. I.27 Let ⟨a_{n}⟩ and ⟨b_{n}⟩ be sequences in [−∞, ∞]. Show

(a) lim sup a_{n} = − lim inf(−a_{n})

(b) lim inf(a_{n} + b_{n}) ≥ lim inf a_{n} + lim inf b_{n}

I.28 Show every open subset U of ℝ can be written has a countable disjoint union

∪_{i∈I}(a_{i}, b_{i}) of disjoint open intervals. I.29 Let E be a subset of ℝ^{d}. Recall a subset S of E is said to be dense in E if for

each x ∈ E and each ϵ > 0, one has B_{ϵ}(x) ∩ S ≠ ∅. Using the countability of the set of points in ℝ^{d} with rational entries, show every subset E of ℝ^{d} has a countable dense subset.

I.30 Let F_{1} and F_{2} be disjoint compact subsets of ℝ^{d}. Show d(F_{1}, F_{2}) > 0.

I.31 Show if F_{1} and F_{2} are nonempty compact subsets of ℝ^{d}, there are points x_{1} ∈ F_{1}

and x_{2} ∈ F_{2} with d(x_{1}, x_{2}) = d(F_{1}, F_{2}). I.32 Show if F ⊆ O ⊆ ℝ^{d} and F is compact and O is open, then there is a continuous

function f such that 0 ≤ f(x) ≤ 1 for all x ∈ ℝ^{d} and f(x) = 1 on F and f(x) = 0 for x ∉ O. Hint: Consider f(x) = min \{\frac{d(x,O^{c})}{d(F,O^{c})}, 1\}.

I.33 Assume that A ≥ 0, B > 0, and f is continuous and nonnegative on [a, b]. Assume that f(t) ≤ A + B ∫_{a}^{t} f(s) ds for a ≤ t ≤ b. Prove that f(t) ≤ Ae^{B(t−a)} for a ≤ t ≤ b.

I.34 Let f : [a, b] → ℝ be monotonically increasing. Show the set of points in [a, b] where f is discontinuous is countable.

2 Lebesgue measure, σ-algebras, Borel sets, measurable functions

Problems concerning ℝ, on ℝ^{n} refer to Lebesgue measure, l. Other problems are about arbitrary measure spaces (X, 𝔄, μ).

II.1 Let A be a σ-algebra of subsets of 𝕏, and let f : 𝕏 → 𝕏 be arbitrary. Prove

that the set

{f^{−1}[S] | S ∈ A}

is a σ-algebra.

II.2 Suppose that g : X → ℝ is μ-measurable and that f : ℝ → ℝ is Borel measurable, meaning that f^{−1}[(a, ∞)] is a Borel set for each real number a. Prove that f ∘ g is μ-measurable.

II.3 Prove that every Lebesgue measurable subset of ℝ is sandwiched between an

F_{σ}-set and a G_{δ}-set of the same Lebesgue measure.

II.4  Prove that the Cantor set is a Borel set.

II.5 (a) Show the set of irrational numbers is a G_{δ} set but is not an F_{σ} set. Hint: Show ℚ is not a G_{δ}, for otherwise you could obtain a decreasing sequence G_{n} of dense open sets that have empty intersection. Then use the decomposition of each G_{n} into a disjoint countable union of open intervals.

(b) Using the fact that the set of rational numbers in any closed interval a ≤ x ≤ b where a < b is not a G_{δ} set, give an example of a Borel subset of ℝ which is neither an F_{σ} or a G_{δ} set.

(c) Let f be any function from ℝ to ℝ. Prove that the set of points of discontinuity of f is of type F_{σ}.

(d) Can a function from ℝ to ℝ be continuous on the rationals and discontinuous on the irrationals? What if the roles of the rationals and irrationals are interchanged?

II.6 Let f : ℝ → ℝ be a measurable function that is not almost everywhere infinite. Prove that there exists a subset S ⊂ ℝ of positive measure such that f is bounded on S.

II.7 Let f : X → ℝ^{∗} be an extended real-valued measurable function on a finite measure space (X, 𝔄, μ). Suppose that f(x) is finite for almost all x. Prove that for each ϵ > 0 there exists a set A ∈ 𝔄, with μ(X \ A) < ϵ, such that f is bounded on A. Give an example for which it is impossible to replace A by a μ-null set.

II.8 If E is a measurable subset of [0, 1] prove that there is a measurable subset A ⊂ E such that l(A) = \frac{1}{2}l(E).

II.9 Let E ⊂ ℝ be a measurable set with the property that

l(E ∩ I) ≤ \frac{l(I)}{2},

for every open interval I. Prove that l(E) = 0. II.10 Let E be a measurable set in [0, 1] and let c > 0. If l(E ∩ I) ≥ cl(I), for all open

intervals I ⊂ [0, 1] show that l(E) = 1. II.11 Let A ⊂ (0, 1) be a measurable set and l(A) = 0. Show that

l\{x^{2} : x ∈ A\} = 0 and l\{\sqrt{x} : x ∈ A\} = 0.

II.12 Show the Cantor ternary set is totally disconnected; that is show it contains no nonempty open interval.

II.13 Show the Cantor ternary set consists of all x in [0, 1] that can be written in form

x = ∑_{k=1}^{∞} \frac{a_{k}}{3^{k}}

where each a_{k} is either 0 or 2. Use the digits a_{n} to define the value of Cantor function ϕ at x.

II.14 Let δ_{1}, δ_{2}, …, δ_{d} be nonzero real numbers. Let E be a subset of ℝ^{d}. Define

ϕ(x_{1}, x_{2}, …, x_{d}) = (δ_{1}x_{1}, δ_{2}x_{2}, …, δ_{d}x_{d}).

Show that the outer measure

l^{∗}(ϕ(E)) = |δ_{1}δ_{2} ⋯ δ_{d}| l^{∗}(E).

Then show E is Lebesgue measurable if and only if ϕ(E) is Lebesgue measurable and then

l(ϕ(E)) = |δ_{1}δ_{2} ⋯ δ_{d}| l(E).

II.15 An extended real-valued function f : X → [−∞, ∞] is measurable if f^{−1}[−∞, α) is measurable for each α ∈ ℝ. Show f^{−1}(α) = {x | f(x) = α} is measurable for α ∈ [−∞, ∞].

II.16 Let (X, 𝔄, μ) be a complete measure space, and define a Carathéodory outer measure,

μ^{∗}(E) = inf {∑_{n∈ℕ} μ(A_{n}) | E ⊆ ⋃_{n∈ℕ} A_{n}, A_{n} ∈ 𝔄 ∀n} ,

for all E ∈ 𝔓(X).

(a) If μ(X) < ∞, show that a set E in the power set 𝔓(X) is measurable if, and only if, for each ϵ > 0 there is a measurable set W with outer measure μ^{∗}(E △ W) < ϵ. (Suggestion: Consider the complete semimetric space structure that can be placed on 𝔄.)

(b) Extend the preceding part to the case of a σ-finite measure space.

II.17 Show that there is a sequence of Lebesgue measurable functions f_{n} : [0, 1] → ℝ that converges pointwise everywhere to a function f having the property that there is not a subset A of [0, 1] of Lebesgue measure 0 such that f_{n} converges uniformly to f on the complementary set [0, 1] \ A. Compare this exercise with the statement of Egoroff’s theorem.

II.18 Let Q = [0, 1]^{d} be the unit cube in ℝ^{d} and suppose f : Q → ℝ is a continuous function. Show the Lebesgue measure of the compact set

Γ = {(x, f(x)) | x ∈ Q}

in ℝ^{d+1} is zero. (Hint: Use uniform continuity.) II.19 Let μ be a finite and finitely additive measure on a field 𝔄 ⊆ 𝔓(X). Prove that

μ is countably additive on 𝔄 if and only if for each decreasing sequence

A_{1} ⊇ A_{2} ⊇ … ⊇ A_{n} ⊇ …

with ⋂_{n=1}^{∞} A_{n} = ∅ we have lim_{n→∞} μ(A_{n}) = 0.

II.20 Prove that Lebesgue measure l is translation-invariant on the real line.

II.21 In the semimetric space (𝔏, d) formed from ([0, 1], 𝔏, l), where l is Lebesgue measure, prove that each of the following sets is dense.

(a) O, the set of all open sets in [0, 1].

(b) K, the set of all closed sets in [0, 1].

II.22 Suppose A and B are measurable subsets of ℝ^{n}, each one of strictly positive but finite measure. Prove that there exists a vector c ∈ ℝ^{n} such that

l((A + c) ∩ B) > 0.

(Hints: Consider the outer measure of A and B. Or, alternatively, consider the convolution 1_{−A} ∗ 1_{B}.)

II.23 Let f : X → ℝ. Prove that f is measurable (meaning that f^{−1}(−∞, a] ∈ L for all a ∈ ℝ) if and only if f^{−1}(B) ∈ L for each Borel set B. (Hint: Show that the family

𝔖 = {A ∈ 𝔓(ℝ) | f^{−1}(A) ∈ L}

is a σ-field.) II.24 Suppose f_{n} : X → ℝ is a measurable function for each n ∈ ℕ, where (X, 𝔄, μ)

is a measure space. Prove that the set

S = {x | lim_{n→∞} f_{n}(x) exists}

is a measurable set.

II.25 Suppose E_{n} is a sequence of measurable sets in X such that ∑_{n∈ℕ} μ(E_{n}) < ∞. Set F_{N} = ⋃_{n≥N} E_{n}.

(a) Show that

lim_{N→∞} μ(F_{N}) = 0.

(b) One defines lim sup E_{n} by

lim sup_{n→∞} E_{n} = ⋂_{N∈ℕ} F_{N}.

Show that lim sup E_{n} consists of all points p which are in infinitely many of the sets E_{n}.

(c) Part (a) shows that μ(lim sup E_{n}) = 0 if ∑_{n=1}^{∞} μ(E_{n}) < ∞. Give an example of a sequence of measurable sets E_{n} such that μ(E_{n}) → 0 as n → ∞, but μ(lim sup E_{n}) ≠ 0.

II.26 Suppose f_{n} is a sequence of real-valued Lebesgue measurable functions on X. Suppose that for each ϵ > 0, μ{x | |f_{n}(x)| ≥ ϵ} → 0 as n → ∞. Prove the following conclusions.

(a) There is a subsequence f_{n_{k}} of f_{n} such that f_{n_{k}}(x) → 0 as k → ∞ almost everywhere.

(b) Show for each ϵ > 0, there is a measurable subset E_{ϵ} of X and a subsequence f_{n_{k}} such that μ(E_{ϵ}) < ϵ and f_{n_{k}}converges uniformly to 0 off E_{ϵ}.

II.27 Prove or give a counterexample: E × F is μ ⊗ ν-measurable in (X, 𝔄, μ) ⨂(Y, 𝔅, ν) if and only if both E and F are measurable with respect to μ and ν respectively.

II.28 Let f ≥ 0 be a Lebesgue measurable function on ℝ^{d}. Define μ(E) = ∫_{E} f dl.

(a) Show that μ is a measure on the Lebesgue measurable sets.

(b) Show that μ ≺ l.

(c) Show that μ is purely infinite on the set E = {x | f(x) = ∞}. (Note μ is purely infinite on E if and only if μ(F) = 0 or μ(F) = ∞ on any measurable subset F of E.)

(d) Show that μ is σ-finite on the set E = {x | f(x) < ∞}. Note this means there is a sequence E_{n} of measurable sets with E = ⋃_{n=1}^{∞} E_{n} and μ(E_{n}) < ∞ for all n.

II.29 It is known the ball of radius 1 in ℝ^{2m} has volume (i.e., Lebesgue measure)

\frac{π^{m}}{m!} .

Find the volume of any ball in ℝ^{2m} which has radius r > 0. II.30 Let f : ℝ^{n} → ℝ be Lebesgue measurable. Prove that f is equal almost every-

where to a Borel measurable function h. Hint: The function f is the pointwise limit of a sequence ϕ_{n} ∈ 𝔖, the set of simple functions. Modify the functions ϕ_{n} suitably.

II.31 Let F be a measurable subset of ℝ. Let E be the set of x in F such that there

is a δ > 0 with m((x − δ, x + δ) ∩ F) = 0. Show that E has measure 0.

II.32 Show there are no countably infinite σ-algebras.

3 The Integral: Convergence Theorems, Product Measures & Fubini’s Theorem.

Integrals on ℝ or ℝ^{n} refer to Lebesgue measure unless stated explicitly to the contrary. Other problems concern integrals on an arbitrary measure space (X, 𝔄, μ).

III.1 Let χ_{[−n,n]}(·) denote the characteristic function of the interval [−n, n] (n ∈ ℕ). Consider the sequence of functions f_{n}(x) := χ_{[−n,n]}(x) sin(\frac{πx}{n}) (x ∈ ℝ).

(a) Determine f(x) = lim_{n→∞} f_{n}(x) and show that the sequence (f_{n})_{n∈ℕ} converges uniformly on compact subsets of ℝ. Does the sequence converge uniformly on ℝ?

(b) Show that

∫_{−∞}^{∞} f(x) dx = lim_{n→∞} ∫_{−∞}^{∞} f_{n}(x) dx.

Are the assumptions of Lebesgue’s dominated convergence theorem satisfied?

III.2 Let f be a positive function on (0, 1] such that f is Riemann integrable on [t, 1] for all t ∈ (0, 1), but lim_{x→0^{+}} f(x) = ∞. Assume that the improper (Riemann) integral (R) ∫_{0}^{1} f(x)dx exists. Show that f is a measurable Lebesgue integrable function and that

∫_{[0,1]} f(x) dx = (R) ∫_{0}^{1} f(x) dx.

III.3 For each of the following problems, check whether the limit exists. If so, find its value.

(a) lim_{n→∞} ∫_{1}^{n}(1 − \frac{x}{n})^{n} dx,

(b) lim_{n→∞} ∫_{1}^{2n}(1 − \frac{x}{n})^{n} dx.

III.4 (a) Characterize in terms of the set of points of continuity those bounded functions on [0, 1] that are Riemann integrable.

(b) Define g on [0, 1] by

g(x) = \begin{cases}0 & if x is irrational or 0, \\ 1/q & if x = p/q in lowest terms. \end{cases}

Is g a Riemann integrable function? Give a proof of your assertion.

III.5 Show that if

f(x) = \begin{cases}1 & if if x is rational, \\ −1 & if if x is irrational, \end{cases}

then f is not Riemann integrable on the interval [0, 1]. Is f Lebesgue integrable? Explain.

III.6 Show there are no bounded sequences (a_{n})_{n∈ℕ} and (b_{n})_{n∈ℕ} for which

f_{n}(x) = a_{n} sin(2πnx) + b_{n} cos(2πnx)

converges to 1 almost everywhere on [0, 1]. III.7 Let f(x) be a real-valued measurable function on a finite measure space

(X, 𝔄, μ). Show that

lim_{n→∞} ∫_{X} cos^{2n}(πf(x)) dμ = μ{x | f(x) ∈ ℤ},

where ℤ is the set of integers. III.8 (a) Show that f(x) = x^{−r} is a Lebesgue integrable function on [0, 1] if 0 ≤

r < 1.

(b) If 0 ≤ r < 1 let

a_{n} = ∫_{0}^{1} \frac{dx}{n^{−1} + x^{r}} .

Compute lim_{n→∞} a_{n}. Be sure to quote the appropriate integration theorems to justify your calculations.

III.9 Let f_{n} : ℝ → ℝ be defined by

f_{n}(x) = \begin{cases}\frac{1}{n} & if |x| ≤ n, \\ 0 & if |x| > n. \end{cases}

(a) Show that f_{n} converges to 0 uniformly on ℝ, and that

lim_{n→∞} ∫_{ℝ} f_{n}(x) dx = 2

while

∫_{ℝ} ( lim_{n→∞} f_{n}) (x) dx ≠ 2.

(b) Explain why this example does not contradict the Lebesgue dominated

convergence theorem.

III.10 (a) Show that f(x) = \frac{1}{\sqrt{x}} is Lebesgue integrable on (0, 1).

(b) Find

inf \{∫_{0}^{1} ψ(x) dx | ψ ∈ 𝔖, ψ(x) ≥ \frac{1}{\sqrt{x}} ∀ x ∈ (0, 1)\} .

(Here 𝔖 is the set of simple functions, which are finite linear combinations of characteristic functions of measurable sets with extended real-valued coefficients.)

III.11 Give an example of a sequence (f_{n})_{n∈ℕ} of bounded, measurable functions on [0, 1) such that

lim_{n→∞} ∫_{0}^{1} |f_{n}(x)| dx = 0

but such that f_{n} converges pointwise nowhere. III.12 Prove that

lim_{n→∞} ∫_{−∞}^{∞}(sin nt)f(t)dt = 0

for every Lebesgue integrable function f on ℝ. (Hint: Do the problem first for step functions.)

III.13 Let

f_{n}(x) = \frac{n}{x(ln x)^{n}}

for x ≥ e and n ∈ ℕ.

(a) For which n ∈ ℕ does the Lebesgue integral ∫_{e}^{∞} f_{n}(x) dx exist?

(b) Determine lim_{n→∞} f_{n}(x) for x > e.

(c) Does the sequence (f_{n})_{n∈ℕ} satisfy the assumptions of Lebesgue’s dominated convergence theorem?

III.14 Define f(x) = ∫_{ℝ} cos(xy) g(y) dy for x ∈ ℝ where g is an integrable function on

ℝ. Show that f is continuous. III.15 Let f be a differentiable function on [−1, 1]. Prove that lim_{ϵ→0} ∫_{ϵ<|x|≤1} \frac{1}{x}f(x) dx

exists. III.16 Let f_{n}(x) = \frac{x^{n}}{n!} e^{−x} for all integers n ≥ 0.

(a) Show that lim_{n→∞} f_{n}(x) = 0 for all x > 0.

(b) Show that f_{n} ∈ L^{1}(0, ∞) with ∥f_{n}∥_{1} = 1 for all n ≥ 0.
(c) Show that lim_{k→∞} ∫_{0}^{k} \frac{x^{n}}{n!} (1 − \frac{x}{k})^{k} dx = 1 for all n ≥ 0.

III.17 Prove that lim_{b→∞} ∫_{0}^{b} \frac{sin x}{x} dx exists but that the function \frac{sin x}{x} is not integrable

over (0, ∞). III.18 Compute the following limit and justify your calculations:

lim_{n→∞} ∫_{0}^{∞} (1 + \frac{x}{n})^{−n} sin (\frac{x}{n}) dx

III.19 Find and justify the limits:

(a) lim_{n→∞} ∫_{0}^{n} \frac{sin x}{1 + nx^{2}} dx

(b) lim_{n→∞} ∫_{0}^{e^{n}} \frac{x}{1 + nx^{2}} dx.

III.20 Let f ∈ L^{1}(0, ∞), and suppose that ∫_{0}^{∞} x|f(x)| dx < ∞. Prove that the function

g(y) = ∫_{0}^{∞} e^{−xy}f(x)dx

is differentiable at every y ∈ (0, ∞). III.21 Let f ∈ L^{1}(ℝ) with respect to Lebesgue measure, and suppose that

∫_{ℝ} |x||f(x)| dx < ∞.

Show that the function

g(y) = ∫_{ℝ} e^{ixy}f(x)dx

is differentiable at every y ∈ ℝ. III.22 Prove that, if f is a real-valued Lebesgue integrable function on ℝ, then

lim_{x→0} ∫_{ℝ} |f(x + t) − f(t)| dt = 0.

III.23 Let f ∈ L^{1}(X, 𝔄, μ), a σ-finite measure space with X = ˙⋃_{n∈ℕ} X_{n} such that μ(X_{n}) < ∞ for each n. Let A_{n} ∈ 𝔄 be such that μ(A_{n}) → 0.

(a) Prove: lim_{n→∞} ∫_{A_{n}} f dμ = 0.

(b) Prove or disprove: lim_{N→∞} ∫_{⋃_{n≥N} X_{n}} f(x) dx = 0.

III.24 Give an example of a sequence of uniformly bounded measurable functions f_{n} on [0, 1] that converges in measure to 0 as n → ∞, yet the sequence f_{n}(x) does not converge for any x ∈ [0, 1].

III.25 For each n ≥ 3 let

f_{n}(x) = \begin{cases}n^{2}x, & 0 ≤ x < \frac{1}{n} \\ 2n − n^{2}x, & \frac{1}{n} ≤ x ≤ \frac{2}{n} \\ 0, & \frac{2}{n} < x ≤ 1. \end{cases}

Sketch the graphs of f_{3} and f_{4}. Prove that if g is a continuous real-valued function on [0, 1], then

lim_{n→∞} ∫_{0}^{1} f_{n}(x)g(x)dx = g(0).

(Hint: First show that ∫_{0}^{1} f_{n}(x) dx = 1.)

III.26 Assume that the real-valued measurable function f(t, x) and its partial derivative \frac{∂}{∂t}f(t, x) are bounded on [0, 1]^{2}. Show that for t ∈ (0, 1)

\frac{d}{dt} [∫_{0}^{1} f(t, x) dx] = ∫_{0}^{1} \frac{∂}{∂t}f(t, x) dx.

(Hint: Consider the difference quotient for the derivative on the left.) III.27 Prove that if f_{n} is Lebesgue integrable on [0, 1] for each n ∈ ℕ, and

∑_{n∈ℕ} ∫_{0}^{1} |f_{n}(x)| dx < ∞,

then ∑_{n∈ℕ} f_{n}(x) is convergent almost everywhere, and

∫_{0}^{1} ∑_{n∈ℕ} f_{n}(x) dx = ∑_{n∈ℕ} ∫_{0}^{1} f_{n}(x)dx.

III.28 Let f ∈ L^{1}(ℝ). Prove that lim_{n→∞} ∫_{0}^{n} e^{−nx}f(x) dx = 0.

III.29 Let f be a nonnegative integrable function on [0, 1]. Suppose that for every n ∈ ℕ

∫_{0}^{1}(f(x))^{n} dx = ∫_{0}^{1} f(x) dx.

Show that f is almost everywhere the characteristic function for some measurable set.

III.30 Provide an example of a sequence {f_{n}} of measurable functions on [0, 1] such that f_{n} → f almost everywhere, and f_{n} ≥ 0, yet lim inf ∫_{0}^{1} f_{n} dl ≠ ∫_{0}^{1} f dl.

III.31 Determine

lim_{n→∞} ∫_{1}^{∞} sin (\frac{x}{n}) \frac{n^{3}}{1 + n^{2}x^{3}} dx.

III.32 Give an example of a sequence of measurable real-valued functions f_{n} such that

lim inf ∫ f_{n}(x) dx < ∫ lim inf f_{n}(x) dx.

Also give a sequence where f_{n} ≥ 0 and

∫ lim inf f_{n}(x) dx < lim inf ∫ f_{n}(x) dx.

III.33 Prove the function

ϕ(x) = ∫_{0}^{∞} \frac{t^{x−1}}{1 + t} dt

is continuous for 0 < x < 1. Is ϕ(x) differentiable on 0 < x < 1? III.34 Let f be a real-valued function on ℝ × ℝ. Suppose that

x ↦ f(x, y)

is continuous for each y, and that

y ↦ f(x, y)

is Lebesgue measurable for each x. Show f is Lebesgue measurable as a function on the plane. (Hint: Express f as a pointwise limit of measurable functions on the plane.)

III.35 Let p(x_{1}, x_{2}) be a polynomial in two variables, but not the zero polynomial. Prove that the set of points x ∈ ℝ^{2} with p(x) = 0 has 2-dimensional Lebesgue measure (l × l) (p^{−1}(0)) = 0.

III.36 Let (X, 𝔄, μ) be a measure space, where 𝔄 is the σ-field generated by a field 𝔈 and μ is the countably additive measure arising via the E. Hopf Extension theorem from a finitely additive measure on 𝔈. Suppose f is integrable on this measure space and that ∫_{E} f dμ = 0 for each E ∈ 𝔈. Show that f(x) = 0 μ-almost everywhere.

III.37 Prove that the family S of step-functions having finite carrier on the real line is

dense in L^{1}(ℝ). III.38 Suppose f ∈ L^{1}(X, 𝔄, μ). Show if ∫_{A} f dμ ≥ 0 for all μ-measurable sets A, then

f(x) ≥ 0 for almost all x. III.39 Show if f is an integrable function on ℝ^{d} and one has

∫_{ℝ^{d}} fh dl = 0

for every bounded continuous function h on ℝ^{d}, then f = 0 almost everywhere. III.40 Let f be a Lebesgue integrable function on ℝ and let E be a bounded measurable

set. Let x_{n} be a sequence in ℝ with |x_{n}| → ∞ as n → ∞. Show

∫_{x_{n}+E} f dl → 0

as n → ∞. III.41 Let f be a real-valued measurable function on a complete measure space

(X, 𝔄, μ). If g = f almost everywhere, prove that g is measurable.

The following three problems are related, but each has sufficient substance to stand alone as a problem.

III.42 Suppose h : ℝ × ℝ → ℝ is given by h(x, y) = x − y. Let N ⊂ ℝ be both a Borel set and a Lebesgue null set. Prove that h^{−1}(N) is a null set in ℝ^{2}. (Hint: Use Fubini’s Theorem. Be sure to justify that your integrand for Fubini’s Theorem is measurable.)

III.43 Let g : ℝ^{2} → ℝ be Borel measurable and ϕ : ℝ → ℝ be Lebesgue measurable. Show that ϕ∘g is Lebesgue measurable on ℝ^{2}, provided that g^{−1} carries Lebesgue null sets to Lebesgue null sets.

III.44 Suppose f and h are in L^{1} (ℝ), and let H(x, y) = f(x − y)h(y). Show that H ∈ L^{1} (ℝ^{2}). Use this to show the function

f ∗ h(x) = ∫_{ℝ} f(x − y)h(y) dy

is defined almost everywhere and is an integrable function on ℝ. Then show that ∥f ∗ h∥_{1} ≤ ∥f∥_{1}∥h∥_{1}. (Hint: Use Fubini’s Theorem. You may assume the measurability of the integrand of the double integral over the plane.)

III.45 Let f : [0, 1] → ℝ be Lebesgue measurable and suppose H(x, y) = f(x) − f(y) is integrable on [0, 1]^{2}. Show f is integrable on [0, 1] and determine the integral of H.

III.46 Suppose f is integrable on ℝ^{d}. Prove

H(α) = m{x | f(x) > α} − m{x | f(x) < −α}

is integrable for α ≥ 0 and show

∫_{[0,∞)} H dl_{1} = ∫_{ℝ^{d}} f dl.

III.47 Let (X, 𝔄, μ) be a measure space for which μ(X) < ∞. Show

L^{q}(X, 𝔄, μ) ⊆ L^{p}(X, 𝔄, μ)

whenever 1 ≤ p ≤ q ≤ ∞. III.48 Let (X, 𝔄, μ) be a probability space, meaning that the nonnegative measure

μ(X) = 1. Suppose that f ∈ L^{p}(X, 𝔄, μ).

(a) Use Jensen’s inequality to prove that

(∫_{X} |f|^{p} dμ)^{\frac{1}{p}} ≤ (∫_{X} |f|^{q} dμ)^{\frac{1}{q}} ,

provided that 1 ≤ p < q < ∞.

(b) Prove that ∥f∥_{p} ≤ ∥f∥_{q} for all 1 ≤ p ≤ q ≤ ∞.

III.49 Let f ∈ L^{1}(ℝ^{2}). Show ∫_{0}^{1} f(x, y +n) dl(y) exists a.e. in x and defines a function

H_{n} in L^{1}(ℝ). Then determine if the sequence H_{n} has a limit in L^{1}(ℝ). III.50 Let f be defined on a measure space, (X, 𝔄, μ). Suppose that for each ϵ > 0,

there is a set W ∈ 𝔄 such that μ(W) < ϵ and f|_{W^{c}} is measurable on W^{c}. Prove that f is Lebesgue measurable.

4 Bounded Variation, Absolutely Continuous Functions and Measures, Convex Functions, the Fundamental Theorem of Calculus, and the Radon-Nikodym Derivative.

Integrals on ℝ or ℝ^{n} refer to Lebesgue measure. Other problems concern integrals on an arbitrary measure space (X, 𝔄, μ).

IV.1 Let F and G be functions on [a, b] into ℝ and let c be a constant. If F and G have bounded variation, prove that

V_{a}^{b}(F + G) ≤ V_{a}^{b}F + V_{a}^{b}G,

and that V_{a}^{b}cF = |c|V_{a}^{b}F.

IV.2 Show f has bounded variation if and only if both f^{+} and f^{−} have bounded

variation. Hint: Show that |f| has bounded variation. IV.3 Show if f is absolutely continuous on [a, b], then the total variation V_{a}^{x}f where

a ≤ x ≤ b is given by

V_{a}^{x}f = ∫_{a}^{x} |f′(t)| dl(t).

Hint: First show F′(x) ≥ |f′(x)| a.e by using V_{x}^{x+h}f ≥ |f(x + h) − f(x)| and then conclude ∫_{a}^{c} |f′(x)| dx ≤ ∫_{a}^{c} F′(x) dx ≤ V_{a}^{c}f. Then show V_{a}^{c}f ≤ ∫_{a}^{c} |f′(x)| dx.

IV.4 Show f : [a, b] → ℝ is absolutely continuous if and only if f^{+} and f^{−} are

absolutely continuous on [a, b]. IV.5 Let V_{a}^{x}g denote the total variation of g from a to x. Show if f is absolutely

continuous on [a, b], then V_{a}^{x}f = V_{a}^{x}|f| for a ≤ x ≤ b and then give an example of a function f of bounded variation where V_{a}^{x}f and V_{a}^{x}|f| are different. You may use the fact that V_{a}^{x}f = ∫_{a}^{x} |f′| dl if f is absolutely continuous.

IV.6 If f is continuous on an interval [a, b] and has a bounded derivative in (a, b), show that f is of bounded variation on [a, b]. Is the boundedness of f′ necessary for f to be of bounded variation? Justify your answer.

IV.7 Let f(x) = x^{2} sin(\frac{1}{x}), g(x) = x^{2} sin(\frac{1}{x^{2}} ) for x ≠ 0, and f(x) = g(x) = 0 for x = 0.

(a) Show that f and g are differentiable everywhere (including at x = 0).

(b) Show that f is bounded variation on the interval [−1, 1], but g is not.

(c) Let f(x) = x sin(1/x) for x ≠ 0 and f(x) = 0 for x = 0. Is f of bounded

variation on [−1, 1]?

IV.8 Show that the product of two absolutely continuous functions on a closed finite

interval [a, b] is absolutely continuous. IV.9 A real-valued function f on an interval I for which there exists a constant C

such that

|f(x) − f(y)| ≤ C|x − y|

for all x and y in I is called a Lipschitz function.

(a) Show that a Lipschitz function is absolutely continuous.

(b) Show that an absolutely continuous function f on an interval is Lipschitz

if and only if f′ is essentially bounded.

IV.10 Let f be a nonnegative integrable function on [0, 1] and let I = ∫_{0}^{1} f(x) dx. Show that

\sqrt{1 + I^{2}} ≤ ∫_{0}^{1} \sqrt{1 + f^{2}(x)} dx ≤ 1 + I.

IV.11 Suppose f is a nonnegative integrable function on [0, 1]. Prove that

\sqrt{∫_{0}^{1} f(t) dt} ≥ ∫_{0}^{1} \sqrt{f(t)} dt.

IV.12 (a) Provide an example of a function on [0, 1] that is not absolutely continuous but is of bounded variation.

(b) Provide examples of two different continuous functions on [0, 1] that have

the same derivative a.e. and that are both equal to zero at 0.

IV.13 Suppose F is absolutely continuous on [0, 1] and that g ∈ L^{1}([0, 1]), with ∫_{0}^{1} g = 0. Prove the “integration by parts” law:

∫_{0}^{1} F (x) g(x) dx = − ∫_{0}^{1} [F′(x) ∫_{0}^{x}g] dx

IV.14 (a) Provide an example of a function of unbounded variation on [0, 1] that has a derivative equal to zero at almost all x ∈ [0, 1].

(b) Provide an example of a function that is absolutely continuous on [0, 1]

but has an unbounded derivative.

IV.15 Prove that, if f is differentiable a.e. on [0, 1] and f′ is not in L^{1}[0, 1], then f is

not of bounded variation on [0, 1]. IV.16 Suppose that f is a real-valued function of bounded variation on [0, 1]. Prove

that

(a) f has a right- and left-hand limit at each point in (0, 1);

(b) f can have only countable many points of discontinuity;

(c) If, in addition to being of bounded variation on [0, 1], f is absolutely continuous on [0, T] for each T < 1, then there exists an absolutely continuous function g on [0, 1] that coincides with f on [0, 1).

IV.17 Suppose that both f and \frac{∂f}{∂y} lie in L^{1} ([a, b] × [c, d]). Suppose also that f(x, y) is absolutely continuous as a function of y for almost all fixed values of x. Prove that

\frac{∂}{∂y} ∫_{a}^{b} f(x, y) dl(x) = ∫_{a}^{b} \frac{∂f}{∂y} (x, y) dl(x)

for almost all y. Take care to establish that both sides exist. Hint: Use Fubini’s Theorem to prove that

g(y) = ∫_{a}^{b} f(x, y) dl(x) − ∫_{c}^{y} (∫_{a}^{b} \frac{∂f}{∂t} (x, t) dl(x)) dl(t)

is a constant function of y.

IV.18 Suppose that we have two decompositions of the measure space for a signed measure μ, as in the Hahn Decomposition Theorem. That is

X = P ∪ N = P′ ∪ N′,

where each measurable subset of P or P′ has nonnegative measure, and each measurable subset of N or N′ has non-positive measure. Prove that every measurable subset of

(P △ P′) ∪ (N △ N′)

is a μ-null set. IV.19 Suppose that the (nonnegative) measures λ, μ, ν on a σ-field 𝔄 ⊂ 𝔓(X) have

the relationship

λ ≺ μ ≺ ν

where λ and μ are finite and ν is σ-finite. Prove that λ ≺ ν and that

\frac{dλ}{dν} = \frac{dλ}{dμ} \frac{dμ}{dν}.

(Hints: Let f = \frac{dλ}{dμ}, g = \frac{dμ}{dν}, h = \frac{dλ}{dν} and show that there is a sequence f_{n} of special simple functions such that f_{n} ↗ f pointwise almost everywhere. Show that |λ(A) − ∫_{A} f_{n} dμ| → 0 for all A ∈ 𝔄 as n → ∞. Show that ∫_{A} f_{n} dμ = ∫_{A} f_{n} g dν for all A ∈ 𝔄. Use the essential uniqueness of the Radon-Nikodym derivative to complete the proof that f_{n}g_{n} → h.)

IV.20 Let E_{1} ⊃ E_{2} ⊃ … ⊃ E_{n} ⊃ … be a decreasing nest of measurable sets in the complete measure space (X, 𝔄, μ). Let f be integrable on (X, 𝔄, μ) and suppose

μ ( ⋂_{n∈ℕ} E_{n}) = 0.

Use the Radon-Nikodym derivative to prove that ∫_{E_{n}} f dμ → 0 as n → ∞ by letting ν be defined by \frac{dν}{dμ} = f.

IV.21 We consider the relationship between the concept of absolute continuity of functions on the real line with that of absolute continuity of measures.

(a) If λ and μ are any two finite measures on a σ-field 𝔄 ⊂ 𝔓(X), prove that λ ≺ μ if and only if they satisfy the following condition: for each ϵ > 0 there exists a δ > 0 such that μ(A) < δ implies that λ(A) < ϵ. (Hint: For one direction, use the Radon-Nikodym theorem.)

(b) Suppose now that the finite measure λ is defined on the Lebesgue measurable sets of ([a, b), 𝔏). Define f(x) = λ[a, x) for all x ∈ [a, b]. Prove that f is an absolutely continuous function on [a, b] if and only if λ ≺ l. (Hint: From right to left is easy by part (a). For the other direction, prove that λ(A) = ∫_{A} f′ dl for all A ∈ 𝔏.)

IV.22 Let μ and ν be nonnegative finite measures on (X, 𝔄). If ν ⊥ μ and ν ≺ μ,

prove that ν = 0, the identically zero measure on 𝔄. IV.23 Let f be a monotone increasing function on [a, b] and define a measure μ by

letting it assign to an interval [a, x) the measure μ[a, x) = f(x) − f(a).

(a) Let μ_{1} be the absolutely continuous part of μ with respect to Lebesgue measure, and find the Radon-Nikodym derivative

\frac{dμ_{1}}{dl} .

(b) Show that the singular part μ_{0} and the absolutely continuous part μ_{1} of μ_{f} can be used to define absolutely continuous and singular parts of the function f.

5 L^{p}, L^{2}, Dual Spaces, Topics from either 7311 or 7330.

Integrals on ℝ or ℝ^{n} refer to Lebesgue measure. Other problems concern integrals on an arbitrary measure space (X, 𝔄, μ).

V.1 Let C([a, b]) be the space of real continuous functions on a closed interval [a, b] equipped with the sup norm. Let M = {f ∈ C([a, b]) : f(x) > 0 for all x ∈ [a, b]}. Show that M is an open subset of C([a, b]).

V.2 Let X be the normed linear space obtained by putting the norm ∥f∥_{1} = ∫_{0}^{1} |f(t)| dt on the set of real continuous functions on [0, 1].

(a) Show that X is not a Banach space.

(b) Show that the linear functional Λf = f(1/2) is not bounded.

V.3  Show that L^{p}[0, 1] is separable for 1 ≤ p < ∞, but not separable for p = ∞.

V.4 Consider a Lebesgue measurable function f on ℝ with ∫_{ℝ} f(t)^{2} dt < ∞. Show

that the function

g(x) = ∫_{ℝ} f(t − x)f(t) dt

is continuous. V.5 Show that L^{p}(0, 1) ⊂ L^{q}(0, 1) for any p > q ≥ 1. Here the integrability is with

respect to the Lebesgue measure. Is the inclusion map for L^{p}(0, 1) into L^{q}(0, 1) continuous?

V.6  Prove or disprove the equality L^{∞}[0, 1] = ⋂_{1≤p<∞} L^{p}[0, 1].
V.7  Let f ∈ L_{p}(ℝ) with 1 ≤ p < ∞. Show that ∫_{|x|>n} |f(x)|^{p} dx → 0 for n → ∞.

V.8 Let g_{n} = nχ_{[0,n^{−3}]}. Show that ∫_{0}^{1} f(x)g_{n}(x) dx → 0 as n → ∞ for all f ∈

L^{2}[0, 1], but not all f ∈ L^{1}[0, 1]. V.9 Provide an example of the following:

(a) A nonzero bounded linear functional on L^{p}[0, 1], 1 < p < ∞.

(b) A nonzero bounded linear functional on ℓ^{∞}.

V.10 Describe precisely how the dual of ℓ^{1} is represented concretely.
V.11 Why is the dual of L^{∞} not equal to L^{1}, in other words, why is L^{1} not reflexive?

V.12 What is the completion of the space of continuous functions on [0, 1] in the

p-norm (1 ≤ p < ∞)? In the ∞-norm? V.13 Let p + q = pq. For g ∈ L^{q}(E), define ĝ ∈ (L^{p}(E))^{∗} as ĝ(f) = ∫_{E} gf. Prove

that ∥ĝ∥ = ∥g∥_{L^{q}(E)}.

V.14 Prove that the linear space of finite sequences is dense in ℓ^{p} for 1 ≤ p < ∞, but

it is not dense in ℓ^{∞}.

V.15 Prove that L^{∞}(ℝ) ∩ L^{p}(ℝ) is dense in L^{p}(ℝ).
V.16 Let 1 < p < q < r < ∞. If f ∈ L^{p}(ℝ) and f ∈ L^{r}(ℝ), then f ∈ L^{q}(ℝ).
V.17 Prove that ℓ^{∞} is not separable, that is, it has no countable dense set.
V.18 Let f ∈ L^{∞}[0, 1] and ∫_{0}^{1} x^{n}f(x) dx = 0 for n ∈ ℕ. Show that f = 0 a.e.

V.19 Let f be a nonnegative Lebesgue measurable function on (0, ∞) such that f^{2} is integrable. Let F (x) = ∫_{0}^{x} f(t) dt where x > 0. Show that lim_{x→0^{+}} \frac{F (x)}{\sqrt{x}} = 0.

V.20 Let f be a measurable nonnegative function on (X, 𝔄, μ) 0 < μ(X) < ∞. Let

∥f∥_{∞} = sup {M | μ(f^{−1}(M − δ, M)) > 0 ∀δ > 0} .

Show that

lim_{n→∞} (∫_{X} f(x)^{n} dμ(x))^{\frac{1}{n}} = ∥f∥_{∞}.

V.21 Let 1 ≤ p < ∞. Suppose f_{n} ∈ L^{p}([0, 1]), ∥f_{n}∥_{p} ≤ 1, and f_{n} → f almost everywhere.

(a) Show that f ∈ L^{p}([0, 1]) and ∥f∥_{p} ≤ 1.

(b) Let 1 < p < ∞ and g ∈ L^{q}([0, 1]) where \frac{1}{p} + \frac{1}{q} = 1. Prove that

∫_{0}^{1} f_{n}g → ∫_{0}^{1} fg.

(c) Give an example to show that the conclusion in item b would be false if

p = 1.

V.22 Let 1 < p < ∞. Suppose 1 ≤ q ≤ ∞ where q ≠ p. Give an example of a

measurable function f on ℝ such that f ∈ L^{p}(ℝ) but f ∉ L^{q}(ℝ). V.23 Suppose 1 ≤ p ≤ q ≤ ∞ and \frac{1}{p} + \frac{1}{q} = 1.

(a) Let f ∈ L^{p}(ℝ) where 1 ≤ p < ∞. Find a function g ∈ L^{q}(ℝ) so that |g|_{q} = 1 and ∫ fg dm = |f|_{p}.

(b) Give an example of an f ∈ L^{∞}(ℝ) for which there is no g ∈ L^{1}(ℝ) such that |g|_{1} = 1 and ∫ fg dm = |f|_{∞}.

V.24 Let f and g be functions in L^{2}(ℝ^{d}). Suppose ∥x_{n}∥ → ∞ as n → ∞. Show

∫ f(x)g(x + x_{n}) dl(x) → 0 as n → ∞.

V.25 Let 1 ≤ p < ∞. Suppose f_{n} ∈ L^{p}(ℝ) and f ∈ L^{p}(ℝ) and

∥f_{n} − f∥_{p} → 0 as n → ∞.

Show that there is a subsequence f_{n(k)} such that f_{n(k)}(x) converges a.e. to f(x) as n → ∞.

V.26 A function f : [0, 1] → L^{1}[0, 1] is called Lipschitz continuous if there exists M > 0 such that ∥f(t) − f(s)∥_{1} ≤ M|t − s| for all t, s ∈ [0, 1]. It is called

differentiable at a point s ∈ (0, 1) if the quotients \frac{f(t) − f(s)}{t − s} converge in L^{1}[0, 1]

as t → s. Let f : [0, 1] → L^{1}[0, 1] be given by f(t) = χ_{[0,t]}, where χ_{[0,t]} denotes the characteristic function of the interval [0, t]. Show that f is Lipschitz continuous and nowhere differentiable.