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This graduate-level course covers Lebesgue’s integration theory with applications to analysis, including an introduction to convolution and the Fourier transform.

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Massachusetts Institute of Technology

November 4, 2022

UK

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Last updated:

November 4, 2022

Duration:

Unlimited Duration

FREE

This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration

Description

This graduate-level course covers Lebesgue’s integration theory with applications to analysis, including an introduction to convolution and the Fourier transform.

Course Curriculum

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

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Massachusetts Institute of Technology