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Browse materialsMIT 18.125 · Fall 2003

Measure and Integration

Art: Anonymous painter,15th century, Westerner and Arab practicing geometry 15th century manuscript, 15th… · Public domain

Instructor: Prof. Jeff Viaclovsky
Term: Fall 2003
Level: Graduate
What's inside: 1 lecture note · 24 other files · 3 web pages
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Source: MIT OpenCourseWare 18.125 Measure and Integration, Fall 2003, Prof. Jeff Viaclovsky — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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Lecture Notes

24 files

MIT OpenCourseWare, Prof. Jeff Viaclovsky, 18.125 Measure and Integration, Fall 2003.

License: CC BY-NC-SA 4.0 · Edition: Fall 2003 · Open the original course on OCW ↗

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Measure and Integration

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Materials by section

Lecture Notes

Lecture 1: Why Measure Theory? Measure Spaces and Sigma-algebras Operations on Measurable Functions (Sums, Products, Composition) Borel Sets

Lecture 2: Real-valued Measurable Functions Limits of Measurable Functions Simple Functions Positive Measures Definition of Lebesgue Integral

Lecture 3: Riemann Integral Riemann Integrable <-> Continuous Almost Everywhere Comparison of Lebesgue and Riemann Integrals Properties of Positive Measures Elementary Properties of the Lebesgue Integral

Lecture 4: Integral is Additive for Simple Functions Monotone Convergence Theorem Integral is Additive for All Non-negative Measurable Functions Interchanging Summation and Integration Fatou’s Lemma

Lecture 5: Integral of Complex Functions Dominated Convergence Theorem Sets of Measure Zero Completion of a Sigma-algebra

Lecture 6: Lebesgue Measure on Rⁿ Measure of Special Rectangles Measure of Special Polygons Measure of Open Sets (Approximate from within by Polygons) Measure of Compact Sets (Approximate from outside by Opens) Outer and Inner Measures

Lecture 7: Definition of Lebesgue Measurable for Sets with Finite Outer Measure Remove Restriction of Finite Outer Measure (Rⁿ, L, Lambda) is a Measure Space, i.e., L is a Sigma-algebra, and Lambda is a Measure

Lecture 8: Caratheodory Criterion Cantor Set There exist (many) Lebesgue measurable sets which are not Borel measurable

Lecture 9: Invariance of Lebesgue Measure under Translations and Dilations A Non-measurable Set Invariance under Rotations

Lecture 10: Integration as a Linear Functional Riesz Representation Theorem for Positive Linear Functionals Lebesgue Integral is the “Completion” of the Riemann Integral

Lecture 11: Lusin’s Theorem (Measurable Functions are nearly continuous) Vitali-Caratheodory Theorem

Lecture 12: Approximation of Measurable Functions by Continuous Functions Convergence Almost Everywhere Integral Convergence Theorems Valid for Almost Everywhere Convergence Countable Additivity of the Integral

Lecture 13: Egoroff’s Theorem (Pointwise Convergence is nearly uniform) Convergence in Measure Converge Almost Everywhere -> Converges in Measure Converge in Measure -> Some Subsequence Converges Almost Everywhere Dominated Convergence Theorem Holds for Convergence in Measure

Lecture 14: Convex Functions Jensen’s Inequality Hölder and Minkowski Inequalities

Lecture 15: L^p Spaces, 1 Leq p Leq Infty Normed Spaces, Banach Spaces Riesz-Fischer Theorem (L^p is complete)

Lecture 16: C_c Dense in L^p, 1 Leq p < Infty C_c Dense in C_o (Functions which vanish at Infty)

Lecture 17: Inclusions between L^p Spaces? l^p Spaces? Local L^p Spaces Convexity Properties of L^p-norm Smooth Functions Dense in L^p

Lecture 18: Fubini’s Theorem in Rⁿ for Non-negative Functions

Lecture 19: Fubini’s Theorem in Rⁿ for L¹ Functions The Product Measure for Products of General Measure Spaces

Lecture 20: Fubini’s Theorem for Product Measure Completion of Product Measures Convolutions

Lecture 21: Young’s Inequality Mollifiers C^{Infty} Dense in L^p, 1 Leq p < Infty

Lecture 22: Fundamental Theorem of Calculus for Lebesgue Integral Vitali Covering Theorem Maximal Function f in L¹ -> Mf in Weak L¹ (Hardy-Littlewood Theorem)

Lecture 23: Lebesgue’s Differentiation Theorem The Lebesue Set of an L¹ Function Fundamental Theorem of Calculus I

Lecture 24: Generalized Minkowski Inequality Another Proof of Young’s Inequality Distribution Functions Marcinkiewicz Interpolation: Maximal Operator Maps L^p to L^p for 1 < p Leq Infty