Measure and Integration : 18.125 (Fall 2003)

Lectures

Textbooks

Examinations and Homework

There will be homework assignments (scheduled to be determined by a stochastic process) and no exams.

• HW #1, due Tuesday, September 30: Rudin Chapter 1: 3-12.

• HW #2, due Tuesday, October 14: Rudin Chapter 2: 5-9, 20.

• HW #3, due Thursday, November 13: Rudin Chapter 3: 4c, 4d, 10, 11, 13, 18b, 23, 25.

Brief Lecture notes

• Lecture 1: September 4
  
  1. Why measure theory?
  2. Measure spaces and \sigma-algebras.
  3. Operations on measurable functions (sums, products, composition).
  4. Borel sets.
• Lecture 2: September 9
  
  1. Real-valued measurable functions.
  2. Limits of measurable functions.
  3. Simple functions.
  4. Positive measures.
  5. Definition of Lebesgue integral.
• Lecture 3: September 11
  
  1. Riemann integral.
  2. Riemann integrable <-> continuous almost everywhere.
  3. Comparison of Lebesgue and Riemann integrals.
  4. Properties of Positive measures.
  5. Elementary properties of the Lebesgue integral.
• Lecture 4: September 16
  
  1. Integral is additive for simple functions.
  2. Monotone convergence theorem.
  3. Integral is additive for all nonnegative measurable functions.
  4. Interchanging summation and integration.
  5. Fatou's Lemma.
• Lecture 5: September 18
  
  1. Integral of complex functions.
  2. Dominated convergence theorem.
  3. Sets of measure zero.
  4. Completion of a \sigma-algebra.
• Lecture 6: September 23
  
  1. Lebesgue measure on R^n.
  2. Measure of special rectangles
  3. Measure of special polygons.
  4. Measure of open sets (approximate from within by polygons).
  5. Measure of compact sets (approximate from outside by opens).
  6. Outer and inner measures.
• Lecture 7: September 25
  
  1. Definition of Lebesgue measurable for sets with finite outer measure.
  2. Remove restriction of finite outer measure.
  3. ( R^n, L, \lambda) is a measure space, i.e., L is a \sigma-algebra, and \lambda is a measure.
• Lecture 8: September 30
  
  1. Caratheodory criterion.
  2. Cantor Set.
  3. There exist (many) Lebesgue measurable sets which are not Borel measurable.
• Lecture 9: October 2
  
  1. Invariance of Lebesgue measure under translations and dilations.
  2. A nonmeasurable set.
  3. Invariance under rotations.
• Lecture 10: October 7
  
  1. Integration as a linear functional.
  2. Riesz representation theorem for positive linear functionals.
  3. Lebesgue integral is the "completion" of the Riemann integral.
• Lecture 11: October 9
  
  1. Lusin's Theorem (measurable function are nearly continuous).
  2. Vitali-Caratheodory Theorem.
• Lecture 12: October 14
  
  1. Approximation of measurable functions by continuous functions.
  2. Convergence almost everywhere.
  3. Integral convergence theorems valid for almost everywhere convergence.
  4. Countable additivity of the integral.
• Lecture 13: October 16
  
  1. Egoroff's Theorem (pointwise convergence is nearly uniform).
  2. Convergence in measure.
  3. Converge almost everywhere -> converges in measure.
  4. Converge in measure -> some subsequence converges almost everywhere.
  5. Dominated convergence theorem holds for convergence in measure.
• Lecture 14: October 21
  
  1. Convex functions.
  2. Jensen's Inequality.
  3. Holder and Minkowski inequalities.
• Lecture 15: October 23
  
  1. L^p spaces, 1 \leq p \leq \infty.
  2. Normed spaces, Banach spaces.
  3. Riesz-Fischer Theorem (L^p is complete).
• Lecture 16: October 28
  
  1. C_c dense in L^p, 1 \leq p < \infty.
  2. C_c dense in C_o (functions which vanish at \infty).
• Lecture 17: October 30
  
  1. Inclusions between L^p spaces? l^p spaces?
  2. Local L^p spaces.
  3. Convexity properties of L^p-norm.
  4. Smooth functions dense in L^p.
• Lecture 18: November 4
  
  1. Fubini's Theorem in R^n for nonnegative functions.
• Lecture 19: November 6
  
  1. Fubini's Theorem in R^n for L^1 functions.
  2. The product measure for products of general measure spaces.
• Lecture 20: November 13
  
  1. Fubini's Theorem for product measure.
  2. Completion of product measures.
  3. Convolutions.
• Lecture 21: November 18
  
  1. Young's inequality.
  2. Mollifiers.
  3. C^{\infty} dense in L^p, 1 \leq p < \infty.
• Lecture 22: November 20
  
  1. Fundamental Theorem of Calculus for Lebesgue Integral.
  2. Vitali Covering Theorem.
  3. Maximal function.
  4. f \in L^1 -> Mf \in weak L^1 (Hardy-Littlewood Theorem).
• Lecture 23: November 25
  
  1. Lebesgue's Differentiation Theorem.
  2. The Lebesue set of an L^1 function.
  3. Fundamental Theorem of Calculus I.
• Lecture 24: December 9
  
  1. Generalized Minkowski inequality.
  2. Another proof of Young's inequality.
  3. Distribution functions.
  4. Marcinkiewicz interpolation: maximal operator maps L^p to L^p for 1 < p \leq \infty.