Lectures: MWF 1-1:50 pm, 260 Mechanical Engineering Building (starting on Friday August 26)

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 447 or equivalent

Office hours: Tu Dec 13, Wed Dec 14, Th Dec 15, 4-5 pm.

Syllabus (Topics 1-6 are assumed to be known. We shall start with Topic 7)

Grading policy: Comprehensive final exam: 45%

Two midterm exam: 2 X 20% = 40%. First midterm exam: Wed Oct 19, 6-8 pm, Altgeld 143. Second midterm exam: Mon Dec 5, 6-8 pm, Altgeld 347.

Homework: 15%

Final exam: Dec. 16, 7-10 pm, 260 Mechanical Engineering Building.

Course diary:

Aug 24: A review of the Riemann and Darboux integrals. The integral as a positive continuous linear functional on C[a,b].

Aug 26: Boolean algebras and s-algebras.

Aug 29: Topological spaces. Measurable spaces. Continuous functions. Measurable functions.

Aug 31: Properties of real-valued Borel measurable functions.

Sep 2: The extended real line. [-¥,¥]-valued measurable functions. Complex-valued measurable functions. limsup and liminf.

Sep 7: The Baire category theorem.

Sep 9: Applications of Baire's category theorem.

Sep 12: Measure spaces.

Sep 14: General properties of measures. Borel-Cantelli lemmas.

Sep 16: Integration of simple functions. Approximation of measurable functions by simple functions.

Sep 19: Integration of measurable functions.

Sep 21: Lebesgue Monotone Convergence theorem and applications.

Sep 23: Applications of LMC: little Fubini and Fatou's lemma. Integration of complex valued functions.

Sep 26: The "set continuity" of the Lebesgue integral. Lebesgue's Dominated Convergence theorem. Differentiation under the integral.

Sep 28: Connections between Riemann integrability and Lebesgue integrability. A criterion for Riemann integrability of bounded functions.

Sep 30: Outer measure. The s-algebra of Lebesgue measurable sets (Caratheodory's theorem).

Oct 3: The Lebesgue measure as outer measure.

Oct 5: Construction of sets which are not Lebesgue measurable. The Cantor function(s) and sets which are Lebesgue measurable but not Borel measurable.

Oct 7: The structure of Lebesgue measurable sets and Lebesgue measurable functions. Lusin's theorem.

Oct 10: Egorov's theorem. Convergence in measure, first properties.

Oct 12: Cauchy sequences in measure are convergent in measure. L¹ convergence implies convergence a.e. on a subsequence.

Oct 14: Product s-algebras. The product premeasure on the Boolean algebra of elementary sets.

Oct 17: The outer measure of a premeasure. The extension of the product premeasure to the product s-algebra.

Oct 19: The uniqueness of the product of s-finite measures. Monotone classes.

Oct 21: The Fubini-Tonelli theorem. Application to the moments of random variables.

Oct 24: The convolution product. Approximation of Lebesgue measurable sets and integrable functions in Rⁿ.

Oct 26: Functions of bounded variation.

Oct 28: Vitali's covering lemma. Lebesgue's differentiation theorem for monotonic functions.

Oct 31: Lebesgue's differentiation theorem for monotonic functions. II.

Nov 2: Absolute continuous functions.

Nov 4: A weak L¹ estimate for the maximal function.

Nov 7: The second Lebesgue differentiation theorem.

Nov 9: Convex functions. Jensen's inequality.

Nov 11: Applications of Jensen's inequality: AGM, Hölder, Cauchy-Schwarz, and Minkowski's inequalities.

Nov 14: L^p spaces. Definitions and basic properties.

Nov 16: Completeness of L^p spaces. The dual of a normed space. The embedding of L^q in (L^p)^*.

Nov 18: The proof of (L^p)^* = L^q for subintervals of the real line. Some properties of embeddings of L^p spaces.

Nov 28: Abstract Hilbert spaces. Schwarz's inequality. Distance to closed convex subsets in Hilbert spaces.

Nov 30: The orthogonal decomposition. Riesz's representation theorem for bounded functionals on abstract Hilbert spaces.

Dec 2: Bessel's inequality. Characterizations of orthonormal bases. Existence of orthonormal bases.

Dec 5: Hilbertian dimension. Isomorphism of separable Hilbert spaces. Trigonometric series.

Dec 7: The Dirichlet and Fejér kernels. The density of trigonometric polynomials in C(T). A first pointwise convergence result.

Dec 9: More pointwise convergence results. Concluding remarks on the Riesz representation and the Stone-Weierstrass theorem.

HW #1 (due Sep 7)

HW #2 (due Sep 16)

HW #3 (due Sep 28)

HW #4 (due Oct 7)

HW #5 (due Oct 26)

HW #6 (due Nov 2)

HW #7 (due Nov 9)

HW #8 (due Nov 18)

HW #9 (due Dec 02)

HW #10 (due Dec 09)

Last modified: December 16, 2005