MAT 701 Real Variables I

Fall 2018

Textbook

Exams

• Midterm Exam
• Final Exam

Homework

• 3.1: Lebesgue outer measure
• 3.2a: Measurable sets
• 3.2b: Measurable sets
• 3.3: Properties of Lebesgue measure
• 3.4: Measurable sets
• 3.5: Lipschitz transformations
• 3.6: Nonmeasurable sets
• 4.1a: Measurable functions 1}
• 4.1b: Measurable functions 2
• 4.2: Semicontinuous functions (+2.1 Bounded variation)
• 4.3: Egorov and Lusin
• 4.4: Convergence in measure
• 5.1: Integral of nonnegative functions
• 5.2: Properties of the integral of nonnegative functions 1
• 5.2b: Properties of the integral of nonnegative functions 2
• 5.3a: Integral of measurable functions 1
• 5.3b: Integral of measurable functions 2
• 5.4-5: Lebesgue, Riemann, Riemann-Stieltjes
• 6.1: Fubini's theorem
• 6.1-2: Fubini and Tonelli
• 6.3a: Applications of Fubini and Tonelli 1
• 6.3b: Applications of Fubini and Tonelli 2
• 7.2a: Hardy-Littlewood maximal function
• 7.2b: Lebesgue Differentiation Theorem
• 7.4a: Differentiability of monotone functions
• 7.4b: Differentiability of monotone functions 2
• 7.5: Absolutely continuous and singular functions
• 8.1: L^p classes
• 8.2: Hölder and Minkowski
• 8.3: Sequence classes l^p
• 8.4: Banach space properties of L^p and l^p
• 8.5-6-7: Hilbert space properties of L²
• 10.1: Additive set functions and measures
• 10.2: Measurable functions and integration
• 10.3a: Absolute continuous and singular ASF
• 10.3b: Absolute continuous and singular ASF 2