Instructor: Florin Boca

Office: 359 Altgeld Hall

Phone: 244-9928

E-mail: fboca@math.uiuc.edu

Textbook: H.L.Royden, Real Analysis (3rd edition)

Prerequisite: Math 347 or equivalent

Lectures: MWF 12:00-12:50, 347 Altgeld Hall

Office hours: Tu, Th 5-6 or by appointment.

Comp Exam Syllabus (Topics 1-6 are assumed to be known. We'll start with Topic 7)

Grading policy: Comprehensive final exam: 50%

One midterm exam: 30% - two-hour test, scheduled on Friday, March 12, 5:15-7:15 pm, AH 347.

Homework: 20%

Final exam: Tuesday May 11, 7-10 pm, 347 Altgeld

Review sessions: May 6,7,8,9,10 between 6-7 pm (meet in 359 AH) Particular questions will be discussed between 7-8 pm those days.

Course diary:

Jan 21: Introduction: The Riemann integral as a positive continuous linear functional on C[a,b].

Jan 23: Boolean algebras, sigma-algebras. Elementary properties.

Jan 26: Definition of topological spaces and measurable spaces. Existence of sigma-algebras. Borel sets. F_sigma and G_delta sets.

Jan 28: Continuous and measurable functions. Definition and basic properties. A set is measurable if and only if its characteristic function is a measurable map.

Jan 30: The oscillation of a function at a point. Non-trivial examples of measurable sets associated with functions (exercises 53&54 at page 53).

Feb.2: The structure of open sets on the real line. The extended real line. Continuity properties of addition and multiplication.

Feb.4: Open sets in the plane. Elementary properties of measurable functions (essentially the first part of Section 5, p.67 in the book).

Feb.6: Further properties of measurable functions. Simple functions.

Feb.9: Approximation of positive measurable functions by simple functions. Measure spaces (Section 1, p.253).

Feb.11: Basic properties of measure. Borel-Cantelli lemmas.

Feb.13: The definition of Lebesgue integral on a measure space. First properties.

Feb.16: The LMC (Lebesgue monotone convergence theorem).

Feb.18: Consequences of LMC: the additivity of the Lebesgue integral, approximations of the Lebesgue integral, Fubini-type theorem for series.

Feb.20: Fatou's lemma. The integral of a complex-valued function.

Feb.23: The linearity and triangle inequality for integrals of complex-valued functions. The LDC (Lebesgue dominated convergence theorem).

Feb.25: The outer measure on R. The general definition of outer measure and Lebesgue measurable sets.

Feb.27: Measurable Lebesgue sets form a sigma-algebra. Borel sets in R are Lebesgue measurable. The definition of the Lebesgue measure on R.

Mar.1: To be rescheduled.

Mar.3: Equivalent characterizations of Lebesgue measurable sets. Existence of subsets of R which are not Lebesgue measurable.

Mar.5: Cantor sets and functions. Existence of Lebesgue measurable sets which are not Borel.

Mar.8: A comparison between the Riemann and the Lebesgue integrals.

Mar.10: The Lebesgue measure on R^k.

Mar.12: Littlewood's principles. Lusin's theorem.

Mar.15: Finishing the proof of Lusin's theorem. Density of continuous functions with compact support in L^1.

Mar.15, 5:10-6 (make-up class for March 1): Starting Chapter III: Differentiation and Integration. A covering result.

Mar.17: The proof of the covering result. The maximal function and weak L^1 inequalities.

Mar.19: Lebesgue's differentiability theorem. Metric density of a set.

Mar.29: Functions of bounded variation. Vitali's covering lemma.

Mar.31: Functions of bounded variation are differentiable almost everywhere. Definition and first properties of absolutely continuous functions.

Apr.2: Further properties of AC functions.

Apr.5: The fundamental theorem of calculus for AC functions. Baire's category theorem.

Apr.7: Uniform boundedness principle. Convex functions.

Apr.9: Rescheduled on April 14, 5-6 pm.

Apr.12: Jensen's inequality. Holder, Cauchy-Schwarz and Minkowski's inequalities.

Apr.14: Definition of L^p spaces.

Apr.14 (5-6 pm, make-up): Completeness of L^p. Denseness of continuous functions with compact support in L^p.

Apr.16: Dual of Banach spaces. The inclusion of L^q into (L^p)*.

Apr.19: The equality (L^p)*=L^q.

Apr.21: Inner products. Primary properties. Minimizing the distance to a closed convex subset in a Hilbert space./p>

Apr.23: Orthogonal decomposition. The Riesz representation theorem in abstract Hilbert spaces.

Apr.26: Orthonormal and maximal orthonormal sets. Fourier coefficients.

Apr.28: Bessel's inequality. Parseval's identity. Characterizations of maximal orthonormal sets.

Apr.30: Every Hilbert space has an orthonormal basis and is isomorphic to some l^2(A). Integration on product spaces. The Tonelli-Fubini theorem. Application to distribution functions.

May 3: The convolution product on L^1(T). The Dirichlet and Fejer kernels. A first convergence result of trigonometric series.

May 4 (6-7 pm, last lecture): Density of trigonometric polynomials in C(T). Parseval's identity in L^2(T). A tauberian theorem and convergence of trigonometric series.

HW #1 (due Friday Feb 6) - revised (and reduced) version

HW #2 (due Friday Feb 13) - corrected (and reduced) version

HW #3 (due Friday Feb 20) - corrected (and reduced) version

HW #4 (now due Monday March 8)

HW #5 (due Friday March 19)

HW #6 (now due on Monday April 5)

HW #7 (revised version, some changes in problem 3, posted 03/28) (now due Monday April 12)

HW #8 (revised version posted 04/14, now due Monday April 19)

HW #9 (due Friday April 30) - revised 04/24 (a clarification in Problem 4 and some additional hints for Problems 3 and 5)

HW #10 (due Monday May 10 - reduced version, posted 05/02)

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