Metric Spaces and Measure Theory

MATH 73-103 · Fall 2026 · September 14 – November 17, 2026

This course develops two pillars of modern analysis. The first half studies metric spaces as the setting for convergence, continuity, and approximation, building to the Arzelà–Ascoli theorem, the Baire category theorem, and the Banach contraction principle. The second half constructs the general theory of measure and integration. Measurable functions, the abstract Lebesgue integral and its convergence theorems, the Riesz representation theorem, L^p spaces.

Course Information

Class	Monday, Wednesday, Friday, 12:50–1:55 PM
X-hour	Tuesday, 1:20–2:10 PM - Kemeny 343
Office hours	Monday 2:00–3:00 PM · Wednesday 5:00–6:00 PM
Instructor	Mohammad Javad Latifi Jebelli (Mohammad.javad.latifi.jebelli@dartmouth.edu@dartmouth.edu), Kemeny Hall [316]
Prerequisites	Undergraduate real analysis and comfort with rigorous proof.

Textbooks

H. L. Royden & P. M. Fitzpatrick, Real Analysis, 4th ed. — Chapters 9–10 (metric spaces).
Walter Rudin, Real and Complex Analysis, 2nd ed. — Chapters 1, 2, 3, 6 (measure theory).

Course Grade

Component	Weight
Group discussions	10%
Weekly assignments (at-home + in-class)	20%
Midterm	30%
Final exam	40%
Total	100%

The midterm is held in class on Friday, October 16 (covers Weeks 1–5). The final exam is cumulative and takes place during the College's final examination period.

Weekly Plan (tentative)

Week 1: Foundations of metric spaces (Sep 14 – Sep 18)

Examples of metric spaces; open and closed sets, interior and closure; convergent sequences; continuous mappings, uniform continuity, and homeomorphisms. Royden §9.1–9.3

Week 2: Completeness and compactness (Sep 21 – Sep 25)

Cauchy sequences and complete metric spaces; completion of a metric space; total boundedness; compactness and its equivalent characterizations. Royden §9.4–9.5

Week 3: Three fundamental theorems (Sep 28 – Oct 2)

Separable metric spaces; the Arzelà–Ascoli theorem; the Baire category theorem; the Banach contraction principle and its applications. Royden §9.6, §10.1–10.3

Week 4: Measures and measurable functions (Oct 5 – Oct 9)

σ-algebras and measurable spaces; measurable and simple functions; positive measures and their elementary properties. Rudin Ch. 1

Week 5: The abstract Lebesgue integral (Oct 12 – Oct 16)

Integration of nonnegative functions and the Monotone Convergence Theorem. Fatou's lemma. Sets of measure zero. Rudin Ch. 1

Midterm: Friday October 16.

Week 6: Positive Borel measures I (Oct 19 – Oct 23)

integration of complex functions and the Dominated Convergence Theorem. The Riesz representation theorem. Rudin Ch. 1-2

Week 7: Positive Borel measures II (Oct 26 – Oct 30)

Regularity of Borel measures; construction and properties of Lebesgue measure; Lusin's theorem and the Vitali–Carathéodory theorem. Rudin Ch. 2

Week 8: L^p spaces (Nov 2 – Nov 6)

Convex functions; the inequalities of Jensen, Hölder, and Minkowski; the L^p spaces and their completeness; approximation by continuous functions. Rudin Ch. 3

Week 9: Complex measures and duality (Nov 9 – Nov 17)

Total variation and absolute continuity; the Radon–Nikodym theorem and the Lebesgue decomposition; the dual of L^p and the Riesz representation theorem for C₀; review. Rudin Ch. 6

Final exam: during the final examination period.

Course Goals

By the end of the term, students should be able to:

Work rigorously with completeness, compactness, and continuity in metric spaces.
Apply the Arzelà–Ascoli, Baire category, and contraction-mapping theorems.
Construct measures and integrate using the convergence theorems.
Use the Riesz representation theorem and L^p spaces, including Radon–Nikodym and duality.

Accessibility

Students with disabilities who may need disability-related academic adjustments and services for this course are encouraged to see the instructor privately as early in the term as possible. Students requiring such services must consult the Student Accessibility Services office (Carson Hall, Suite 125, 646-9900). All inquiries and discussions will remain confidential.