Measure and Integration
A rigorous course in the modern theory of measure and integration that replaces the Riemann integral with the Lebesgue integral and explains why the change is needed: the failure of Riemann integration for limits such as the Dirichlet function, escaping mass and incompleteness, σ-algebras, Borel sets and measures with continuity from above and below and the π–λ uniqueness theorem, Lebesgue outer measure and Carathéodory’s construction with the proof that intervals have their length and Borel sets are measurable, regularity and Littlewood’s principles, the Cantor set, fat Cantor sets and the Cantor function, a Lebesgue measurable set that is not Borel and Vitali’s non-measurable set, measurable functions, simple-function approximation and Lusin’s theorem, the Lebesgue integral with Markov’s inequality and the monotone convergence theorem, integrable functions, Fatou’s lemma and the dominated convergence theorem with differentiation under the integral sign, Lebesgue’s criterion for Riemann integrability, Thomae’s function and improper integrals, convergence almost everywhere, in measure and in L¹ with Egorov’s theorem, the Borel–Cantelli lemma and Riesz’s subsequence theorem, product measures and the theorems of Tonelli and Fubini with their counterexamples, the layer-cake formula and convolution, the Lᵖ spaces with the inequalities of Young, Hölder and Minkowski and the Riesz–Fischer completeness theorem, and a capstone on the Hardy–Littlewood maximal inequality, the Lebesgue differentiation theorem, absolute continuity, the Lebesgue decomposition and the Radon–Nikodym theorem, and Kolmogorov’s measure-theoretic probability with conditional expectation, with complete proofs, worked examples and runnable Python illustrations throughout.
MATH-441 Year 4 3 credits 12 lessons · 4 h 30 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
What you'll learn
- Explain the limitations of the Riemann integral with respect to limits, convergence theorems and completeness, and describe how slicing the range rather than the domain leads to the Lebesgue integral.
- Work with σ-algebras, generated σ-algebras and Borel sets, prove the basic properties of measures including continuity from above and below, and use the π–λ theorem to show that measures agreeing on a π-system agree on the σ-algebra it generates.
- Construct Lebesgue measure from outer measure by Carathéodory’s criterion, prove that intervals have their length and Borel sets are measurable, and apply regularity to approximate measurable sets by open and closed sets.
- Analyze the Cantor set, fat Cantor sets and the Cantor function, exhibit a Lebesgue measurable set that is not Borel, and prove Vitali’s theorem on the existence of non-measurable sets.
- Prove that measurable functions are closed under algebraic operations and pointwise limits, approximate nonnegative measurable functions by increasing sequences of simple functions, and state Lusin’s theorem.
- Define the Lebesgue integral of nonnegative functions, prove Markov’s inequality and the monotone convergence theorem, and use them to integrate series term by term and define measures by densities.
- Define integrable functions and the space L¹, prove Fatou’s lemma and the dominated convergence theorem, and apply them to series, parameter-dependent integrals and differentiation under the integral sign.
- Prove that Riemann integrable functions are Lebesgue integrable with the same integral, prove Lebesgue’s criterion for Riemann integrability, and compare Lebesgue integrals with improper Riemann integrals.
- Distinguish convergence almost everywhere, almost uniformly, in measure and in L¹, prove the implications among them including Egorov’s theorem and Riesz’s subsequence theorem, and construct counterexamples for the others.
- Construct product measures on σ-finite spaces, prove and apply the theorems of Tonelli and Fubini, and explain the counterexamples that arise without integrability or σ-finiteness.
- Prove the inequalities of Young, Hölder and Minkowski, show that the Lᵖ spaces are complete normed spaces, and describe density results, inclusions and the Hilbert space structure of L².
- Prove the Lebesgue differentiation theorem via the maximal inequality, relate absolute continuity to the fundamental theorem of calculus, state and apply the Radon–Nikodym theorem, and interpret probability and conditional expectation in measure-theoretic terms.
Course content
12 lessons · 4 h 30 min total · final exam
- 1 Why measure theory: the limits of the Riemann integral 24 min · quiz
- 2 σ-algebras and measures 22 min · quiz
- 3 Outer measure and Carathéodory’s construction of Lebesgue measure 24 min · quiz
- 4 Borel sets, Cantor sets and non-measurable sets 22 min · quiz
- 5 Measurable functions and simple-function approximation 21 min · quiz
- 6 The Lebesgue integral and the monotone convergence theorem 21 min · quiz
- 7 Integrable functions, Fatou’s lemma and dominated convergence 22 min · quiz
- 8 Riemann versus Lebesgue integration 21 min · quiz
- 9 Modes of convergence and Egorov’s theorem 21 min · quiz
- 10 Product measures and the theorems of Fubini and Tonelli 23 min · quiz
- 11 Lᵖ spaces: Hölder, Minkowski and completeness 23 min · quiz
- 12 Capstone: differentiation, Radon–Nikodym and probability 26 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A rigorous course in the modern theory of measure and integration that replaces the Riemann integral with the Lebesgue integral and explains why the change is needed: the failure of Riemann integration for limits such as the Dirichlet function, escaping mass and incompleteness, σ-algebras, Borel sets and measures with continuity from above and below and the π–λ uniqueness theorem, Lebesgue outer measure and Carathéodory’s construction with the proof that intervals have their length and Borel sets are measurable, regularity and Littlewood’s principles, the Cantor set, fat Cantor sets and the Cantor function, a Lebesgue measurable set that is not Borel and Vitali’s non-measurable set, measurable functions, simple-function approximation and Lusin’s theorem, the Lebesgue integral with Markov’s inequality and the monotone convergence theorem, integrable functions, Fatou’s lemma and the dominated convergence theorem with differentiation under the integral sign, Lebesgue’s criterion for Riemann integrability, Thomae’s function and improper integrals, convergence almost everywhere, in measure and in L¹ with Egorov’s theorem, the Borel–Cantelli lemma and Riesz’s subsequence theorem, product measures and the theorems of Tonelli and Fubini with their counterexamples, the layer-cake formula and convolution, the Lᵖ spaces with the inequalities of Young, Hölder and Minkowski and the Riesz–Fischer completeness theorem, and a capstone on the Hardy–Littlewood maximal inequality, the Lebesgue differentiation theorem, absolute continuity, the Lebesgue decomposition and the Radon–Nikodym theorem, and Kolmogorov’s measure-theoretic probability with conditional expectation, with complete proofs, worked examples and runnable Python illustrations throughout.
Part of the Science stream, typically taken in year 4.