Topics and HW

Math. 540 (C1): Fall 2009

You must submit home work solutions according to the list of collected problems

Click here to see the lists of collected problems and due dates.

No misplaced hw problems will be accepted

If you have questions connected with grading of homework assignments or exams, please email to Boonyong Srponpaew bsripon2@uiuc.edu

Date	Lectures *.html format (use Internet Explorer only)	Lectures *.pdf format	Chapter: Section	Topics
08.24	1	1	1.1-1.4, 1.7	Algebra of logic. Properties. Sets. The power set. Properties. Infinite operators on sets. Cartesian products. Relations. Reflexivity, symmetry, transitivity and antisymmetry. Mappings. Injective, surjective mappings and bijections. Properties of functions and preimages. Equivalent sets. Power of sets, cardinal number. Countable and infinitely countable sets.
08.26	2	2	Sec. 1.6	Representation of a countably infinite set as a range of a sequence of distinct elements. Countability of infinite subsets of a countably infinite set. Union of a countably infinite set and a finite set. The union of a finite number of pairwise disjoint countably infinite sets. Countably infinite union of pairwise disjoint finite sets. Countably infinite union of pairwise disjoint countably infinite sets. Cartesian product of a finite number of countably infinite sets. Prove that the set of rational numbers is countably infinite. Corollaries. Power of the union of an infinite set and a countably infinite set. Power of an infinite set which is not countably infinite without a countable subset. Non-countability of the segment [0,1]. Corollary for irrational numbers.
08.28	3	3	Sec. 1.6	Power of the set of infinite sequences with natural numbers components. Cardinal number of the union of a finite number, countable, of power of continuum of pairwise disjoint sets of power continuum. Cardinal number of infinite sequences of zeros and ones. Cardinal number of the power set of the set of natural numbers.
08.31	4	4	Sec. 1.6, 1.7	Cantor-Bernstein theorem. Partially ordered sets. An upper bound and maximal element in a partially ordered set. Cardinal numbers and partially ordered sets. Zorn's lemma, Hausdorff maximal principle, well-ordering and the well-ordering principle, the axiom of choice. Start discussion of the Theorem stating that each subset of cardinal numbers is a chain.
09.02	5 Revision: in the definition of the triple injection is replaced with bijection	5 Revision: in the definition of the triple injection is replaced with bijection	1.6, 1.4	Complete the proof of the Theorem stating that each subset of cardinal numbers is a chain. Prove Proposition 8, P. 26 (Royden). Continuum hypothesis. Rings and their properties. Semi-rings. State properties.
09.04	6 Revised	6	1.4	Proof of existence of minimal ring generated by a non-empty family of sets. Examples of rings. Algebras of sets. Proposition 2, Sec. 1.4 (Royden). Semi-rings. Properties. Minimal ring generated by a semi-ring. Theorem on a structure of semi-rings. Examples. Sigma (Borel) and delta-rings. Sigma (Borel) algebras. Sigma algebra of Borel subsets of R^n. F sigma and G delta sets.
09.09	7 Revised	7	2.1-2.6 and 7.1, 7.2, 7.4, 7.8	Review of metric spaces. Open and closed balls in a metric space. Interior points and open sets. Closed sets. Properties of open and closed sets. Interior and closure of a set in a metric space. A set that is dense in another set. Everywhere dense sets. Nowhere dense sets. Convergent of sequences in a metric space. Cauchy sequences. Complete metric spaces. Cantor's theorem on a non-empty intersection of a non-increasing sequence of closed balls in a complete metric space. Sets of the first and the second category. Baire's theorem. Example of a Borel set that is not an F sigma set and of a Borel set that is not a G delta set. Please review Sec. 2.1-2.6 and 7.1, 7.2, 7.4 and 7.8.
09.11	8	8	1.4 and see lectures	Set functions. Additive and sigma-additive set functions. Measures on a semi-ring. Extension of measures from a semi-ring to a ring. Proof that extension is well defined. Example: semi-ring of open half-intervals. Sigma additivity of extended measure from a sigma additive measure on a semi- ring.
09.14	9 Revised	9 Revised	See lectures	Extension to a sigma-additive measure on a ring. Proof of sigma-additivity of the measure m on Lebesgue semi-ring of semi-open intervals [a,b). Sigma-additivity of the measure on the minimal ring generated by Lebesgue's semi-ring. Notion of sigma-sets.
09.16	10 Revised	10 Revised	See lectures and 3.1, 3.2	Countable union of sigma-sets. Example: sigma-set is not a sigma-ring. Sigma sets of a semi-ring and the minimal ring generated by the semi-ring. Extension of the measure on the minimal ring to sigma-sets. Measure of an open interval as a sigma-set. Outer measures. Construction of Lebesgue outer measure on all subsets of R. Proposition on the outer measure of intervals.
09.18	11 Revised	11 Revised	See lectures	Sigma semi-additivity of the Lebesgue outer measure. Outer measure of countable sets. Approximation of the Lebesgue outer measure of a set by Lebesgue outer measure of open set containing given set. Lebesgue outer measure and G. delta sets. Lebesgue-Stieltjes measure on the ring of semi-open intervals [a,b). General property of measure (and sigma-additive measures) on a ring; continuity of a sigma-additive measure. Start discussion of condition for sigma-additivity of Lebesgue-Stieltjes measure.
09.21	12	12	3.3 and see lectures	Continuity of measures in a ring: measure of intersection of a non-increasing sequence of sets. Sigma-additivity of Lebesgue-Stieltjes measures and continuity from the left of the function "fi". Lebesgue measurable sets in the real line. Measurability of R, empty set and sets with zero Lebesgue outer measure. Prove that the set of all Lebesgue measurable sets is an algebra. Abstract "mu"*-measaruble sets. Restriction of a measure to a subset.
09.23	13	13	3.3	Measurability of set w.r.t. restricted outer measure. Lemma 9 (Sec. 3.3.). Proof the theorem stating that Lebesgue measurable sets form a sigma-algebra. Measurability of the interval (a,+infinity). Start Theorem 12 (Sec. 3.3.): measurability of Borel's sets.
09.25	14 Revised	14 Revised	3.3	Representation of open sets in R as disjoint unions of open intervals. Corollary for closed sets. Complete the proof of the measurability of Borel sets. Lebesgue measure. Sigma-semi-additivity. Countable additivity of Lebesgue measure. Continuity of Lebesgue measure: measure of the union of a non-decreasing sequence of Lebesgue measurable sets and measure of the intersection of a non-increasing sequence of Lebesgue measurable sets. Approximation of Lebesgue measurable sets of finite measure by open sets. Statement of Lebesgue measure approximation theorem.
09.28	15	15	3.3	Proposition 15 (Sec. 3.3, Royden) and other approximation results. More on symmetric difference. Symmetric difference as a pseudo-metric. Symmetric difference as the metric on the minimal ring generated by the semi-ring of intervals [a,b).
09.30	16	16	3.3	Metric characterization of bounded Lebesgue measurable sets in terms of sets from minimal ring generated by Lebesgue semi-ring. Cantor set. Characterization of Cantor's set in terms of ternary representations. Cardinal number and Lebesgue measure of Cantor set.
10.02	17	17	3.4 and see lectures	Prove that C is nowhere dense and perfect. Cardinal number of Lebesgue measurable sets. Translation invariance of Lebesgue outer measure and Lebesgue measure. Existence of non-measurable set in [0,1).
10.05		18		Midterm Exam 1. Click here for the program of Exam 1. Click here for Exam 1 solutions. Click here for Exam 1 statistics.
10.07	19 Revised	19 Revised	3.5	Extended -real valued functions. Lebesgue measurable functions. Equivalent definitions. Corollary. Measurability in terms of Borel sets. Start the proof of the theorem on properties of measurable functions.
10.09	20	20	3.5	Continue the proof of the theorem on properties of measurable functions. Borel measurable functions; superposition of Borel measurable and measurable functions, superposition of a continuous and measurable functions. Theorem on sequences of measurable functions: sup, lim, limsup and lim inf. Notion of "almost everywhere". measurability of a limit of measurable functions.
10.12	21 Revised	21 Revised	3.5	Simple, characteristic and step functions, measurability and other properties; f+ and f- functions, measurability. Theorem on approximation by simple functions.
10.14	22 Revised: added: in Egoroff's theorem the set E epsilon can be selected closed See also Step 2 of the proof of Lusin's theorem	22 Revised: added: in Egoroff's theorem the set E epsilon can be selected closed	3.5	Complete the proof of the theorem on approximation by simple functions. Egoroff's theorem. Measurabiluty of functions continuous relative a measurable set. Lusin's C-property. Start the proof of the lemma establishing C-property for simple functions.
10.16	23 Revised: Step 2 of the proof of Lusin's theorem	23 Revised: Step 2 of the proof of Lusin's theorem	3.5	Complete the proof of the lemma establishing C-property for simple functions. Proof of Lusin's theorem.
10.19	24 Revised: second version of Lusin's theorem; typo	24 Revised: second version of Lusin's theorem; typo	3.5	Second version of Lusin's theorem, corollary for f:[a,b]->R. Cantor's function: review of the uniform metric; construction of Cantor's function and proof of main properties, image of C and [0,1]\C.
10.21	25	25	3.5, 4.5	Examples connected with Cantor's function. Convergence in measure. Theorems describing relation between pointwise convergence and convergence in measures. Necessary and sufficient conditions for convergence in measure on sets of finite measure.
10.23	26 Revised	26 Revised	4.5, 4.1	Review of Riemann integral. Upper and lower Darboux sums and integrals. Existence of Riemann integral. Step functions, riemann integral of step-functions. Start the proof of the characterization of the upper and lower Darboux integrals in terms of riemann integrals of step-functions.
10.26	27 Revised	27 Revised	4.1, 4.2	Finish the proof of the characterization of the upper and lower Darboux integrals in terms of riemann integrals of step-functions. Corollary: another definition of Riemann integral. Integral of measurable simple functions vanishing outside a set of finite measure. Calculation of integrals of simple functions in a non-canonical representation.
10.28	28 Revised	28 Revised	4.2	Proof of Propositions 2, 3 (Sec. 4.2, Royden). Criterion of measurability. Lebesgue integral of a bounded measurable function over a measurable set of finite measure. Corollary of Lebesgue integrability of Riemann integrable functions.
10.30	29	29	4.2	Proof of Proposition 5 (Royden). Proof of the bounded convergence theorem. Start Lebesgue's criterion of Riemann integrability.
11.02	30 Revised	30 Revised	4.2, 4.3	Complete the proof of Lebesgue's criterion of Riemann integrability. Integral of non-negative function over a measurable set of infinite measure. Start the proof of Proposition 8, Sec. 4.3, Royden.
11.04	31 Revised	31 Revised	4.3	Complete the proof of Proposition 8, Sec. 4.3, Royden, Chebyshev inequality. Convergence theorems: Fatou's lemma, Monotone convergence theorem, corollary for infinite series, Beppo Levi's theorem.
11.06	32	32	4.3, 4.4.	Non-negative Lebesgue integrable functions. Proof of Propositions 11-14, Royden, Sec. 4.3. General Lebesgue integrable functions. Properties of Lebesgue integrable functions. Proof of Proposition 15, Royden, Sec. 4.4. Integrability of \|f\|. Start Lebesgue dominated convergence theorem.
11.09	33 Revised: corollary to Lebesgue dominated convergence theorem	33 Revised: corollary to Lebesgue dominated convergence theorem	4.4, 5.1	Lebesgue dominated convergence theorem and its corollary. Vitali's covering. Start the proof of Vitali's lemma.
11.11	34	34	5.1	Complete the proof of Vitali's lemma. Start the proof of Lebegue's theorem on differentiability a.e. of a non-decreasing function.
11.13	35	35	5.1	Complete the proof of Lebegue's theorem on differentiability a.e. of a non-decreasing function
11.16		36		Midterm Exam 2. Click here for for the program of Exam 2.
11.18	37 Revised	37 Revised	5.2, 5.3	Total variation, positive and negative parts of the total variation. Functions of bounded variation. Lemma 4, Sec. 5.2 (Royden). Theorem on representation of a function of bounded variation as a difference of two non-decreasing functions. Differentiability a.e. of a function of bounded variation. Lemmas 7, 9 of Sec. 5.3 (Royden). Start the proof of the theorem on differentiability of Lebesgue integral w.r.t. to the upper limit of integration.
11.20	38	38	5.3, 5.4	Complete the proof of the theorem on differentiability of Lebesgue integral w.r.t. to the upper limit of integration. Absolutely continuous functions. Example: Lipschitz functions, Cantor's function as not absolutely continuous. Lemmas 11, 13 and Corollary 12 (Royden, Sec./ 5.4). Absolute continuity and indefinite integrals (Theorem 14, Royden, Sec./ 5.4).
11.30	39	39
12.02	40	40
12.04	41	41
12.07	42	42
12.09	43	43
12.14		Final Exam		8:00-11:00 AM, Monday, December 14, 341 AH.

Collected HW assignments

(Click the assignment number for solutions)


HW Assign.	List of Probllems	Due Date
I	Section 1.1: 3 Section 1.2: 6, 7 Section 1.3: 12, 16 Section 1.6: 24, 25 and click here for one more hw problem	08.31
II	Click here for hw problem (more problems have been added) and solve the following problems: Sec. 1.5: 21	09.09
III	Sec. 1.4: 19 (for part (a), assume that the collection C is non-empty and that your are looking for a smallest sigma-algebra generated by C over the same unit set X; for part (b), first prove that the minimal algebra generated by C consists of sets of C, their finite unions and finite differences; please give all the details of your solution), Sec. 1.4: 20 (assume that C is a family of subsets from the a unit set X). Sec. 1.8: 30 Sec. 1.9: 31	09.14
IV	Click here for hw problems and also solve the following problems Sec. 1.9: 32 Sec. 2.5: 24, 27 (prove the statement of Problem 27 for general metric spaces) Sec. 7.8: 32	09.21
V	Click here for hw problem and also solve the following hw problems: Sec. 2.4: 21 (Hint for part (a): you have to prove that if the sum is finite then E must be countable; try to represent E as a countable union of finite sets by using the assumption that the sum is finite) Sec. 2.5: 28 (Hint: x is in the closure of a set X if and only if for every r>0 the open interval (x-r,x+r) has a non-empty intersection with X), 35 (Hint: consider the collection O of open sets which are complimentary to sets from C without the bounded set; prove that the bounded set cannot be covered by a finite subfamily of the family O, thereby it cannot be covered by O) 36 (and also give a counter-example showing that the statement can be false if no Fn is bounded). Sec. 3.2: 5 (Hint: prove that [0,1]\(union of covering intervals) is either empty or finite).	09.28
VI	Sec. 2.7: 53, 54 (Hint: the sequence converges if and only if it is a Cauchy sequence; so, start with the set Ak(m,n)={x in R \| \|fn(x)-fm(x)\|<=1/k, then use intersections, unions to catch x at which (fn(x )),n=1,2,..., is a Cauchy sequence}. Sec. 3.2: 8 Sec. 3.3: 14b Sec. 3.4: 15, 16, 17	10.12
VII	Click here for hw problems.	10.19
VIII	Sec. 3.5: 28d and click here for more hw problems	10.26
IX	Sec. 4.1: 1a,b and cick here for more hw problems.	11.02
X	Sec. 4.3: 3, 4, 5, 6, 7	11.09
XI	Click here for hw problems and solve the following hw problems: Sec. 4.3: 8 (problem 8 deals with extended-real valued functions, integral of an extended real valued function was discussed in class), 9 Sec. 4.4: 10, 11, 14, 15 Sec. 5.1: 1, 2, 3, 4	11.30
XII		This hw will not be collected

Math. 540 (C1): Fall 2009

Date

Lectures

*.pdf format

Chapter: Section

Topics