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Created by:
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
About the instructor
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Massachusetts Institute of Technology
