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18.125

Measure Theory and Analysis

3-0-9 units · Graduate

Provides a rigorous introduction to Lebesgue's theory of measure and integration. Covers material that is essential in analysis, probability theory, and differential geometry.

Fall 2003 · Measure and Integration

Topics

  • Why Measure Theory?
  • Real-valued Measurable Functions
  • Riemann Integral
  • Integral is Additive for Simple Functions
  • Integral of Complex Functions
  • Lebesgue Measure on R^n
  • Definition of Lebesgue Measurable for Sets with Finite Outer Measure
  • Caratheodory Criterion
  • Invariance of Lebesgue Measure under Translations and Dilations
  • Integration as a Linear Functional
  • Lusin’s Theorem (Measurable Functions are nearly continuous)
  • Approximation of Measurable Functions by Continuous Functions
  • Egoroff’s Theorem (Pointwise Convergence is nearly uniform)
  • Convex Functions
  • L^p Spaces, 1 Leq p Leq Infty
  • C_c Dense in L^p, 1 Leq p < Infty
  • Inclusions between L^p Spaces? l^p Spaces?
  • Fubini’s Theorem in R^n for Non-negative Functions
  • Fubini’s Theorem in R^n for L^1 Functions
  • Fubini’s Theorem for Product Measure
  • Young’s Inequality
  • Fundamental Theorem of Calculus for Lebesgue Integral
  • Lebesgue’s Differentiation Theorem
  • Generalized Minkowski Inequality

Readings

  • Rudin, Walter. Real and Complex Analysis. McGraw-Hill International Editions: Mathematics Series. McGraw-Hill Education - Europe, 1986. ISBN: 9780070542341.
    Required
  • Jones, Frank. Lebesgue Integration on Euclidean Space. Boston: Jones & Bartlett Publishers, February 1, 1993. Evans, Lawrence C., and Ronald F. Gariepy. Measure Theory and Fine Properties of Function. Boca Raton, Florida: CRC Press, December 18, 1991. ISBN: 0849371570.
    Recommended

Prerequisites & course information

Prereq: 18.100A , 18.100B , 18.100P , or 18.100Q G (Spring) 3-0-9 units

Sources