1/11 Th Lecture postponed (due to JMM 2018 San Diego)

1/16 Tu Riemann integral in 1 vble: Darboux definition. Intrgrability of: monotone fns, cont. fns,

fns w/ D_f finite. Sets of measure zero. Lebesgue's characterization of Riemann integrability (statement).

1/18 Th Lebesgue's integrability theorem (proof). Convergence of integrals: examples, theorem on uniform convergence.

HW set 1 (due 1/25)

1/23 Tu Fundamental theorem of Calculus. Cantor set, Lebesgue's function. Lebesgue outer measure.

1/25 Th The problem of measure. Vitali's covering theorem (statement). Dini's derivate numbers.

Lebesgue's theorem on differentiability a.e. of monotone functions (outline of proof.)

Lebesgue's theorem on differentiability a.e. of monotone functions

(includes three problems= HW2, due Feb. 1)

1/30 Tu Functions of bounded variation: basic properties

Functions of bounded variation

2/1 Th Functions of bounded variation (continued)

HW set 3 (due 2/8): exercises 4, 5, 7 and 8 from the handout "functions of bounded variation"

2/6 Tu Rectifiable curves

2/8 Th Absolute continuity

2/13 Tu Riemann-Stieltjes integrals

Problems on Riemann-Stieltjes integrals

(HW 4, due 2/20)

2/15 Th Differential forms and line integrals

Riemann-Stieltjes integrals and line integrals of one-forms (15 pages)

(Includes exercises 1 to 5=HW 5, due Tu 2/27)

2/20 Tu Exterior differential, closed forms, exact forms.

2/22 Th Integration in several variables: continuous functions

2/27 Tu Sard's theorem (equidimensional), Lebesgue's integrability thm: proofs.

3/1 Th: MIDTERM 1. Included: material up to 2/22

Midterm 1 (with solutions)

3/6 Tu Integration over more general sets--Jordan content, J-measurable sets

Jordan Content and Riemann integral

3/8 Th L-Measurable sets in R^n ([Fleming], 5.1 and 5.2)

(due 3/22): Problems 1 and 2 from handout "Jordan content"; problems 1, 3, 12 from [Fleming], p.180/181.

3/13, 3/15: SPRING BREAK

3/20: measurable sets (cont'd)/integral of functions with cpt support (5.10, 5.3)

3/22: integral of bounded measurable functions over bounded measurable sets (5.4)