Measure Theory for Probability
A weekly reading group building measure-theoretic foundations for probability and stochastic processes.
Purpose
This group aimed to build rigorous measure-theoretic foundations needed for modern probability theory and stochastic processes. Targeted at graduate students wanting to move beyond introductory probability into the language of sigma-algebras, Lebesgue integration, and convergence theorems.
Structure
We followed a book-based format, working through Probability and Measure (Billingsley, Anniversary ed.) chapter by chapter. Each week, one member presented the key definitions and theorems, and the group worked through selected problems together.
Session 16 — May 16, 2026
Chapter: Billingsley, Probability and Measure, Chapter 33 — Characteristic Functions
Presenter: Khue
Summary: Final session. We covered characteristic functions and proved the continuity theorem. Wrapped up with a retrospective on the group’s progress over the semester.
Session 15 — May 9, 2026
Chapter: Billingsley, Probability and Measure, Chapter 32 — The Central Limit Theorem
Presenter: Alex
Summary: Proved the Lindeberg–Lévy CLT using characteristic functions. Discussed the Lindeberg and Lyapunov conditions for triangular arrays.
Session 14 — May 2, 2026
Chapter: Billingsley, Probability and Measure, Chapter 31 — The Strong Law of Large Numbers
Presenter: Sam
Summary: Proved the SLLN via truncation and Kolmogorov’s inequality. Compared with the weak law and discussed when the distinction matters in practice.
Session 13 — April 25, 2026
Chapter: Billingsley, Probability and Measure, Chapters 29–30 — Laws of Large Numbers (Weak)
Presenter: Jordan
Summary: Covered Chebyshev’s WLLN and convergence in probability. Discussed the relationship between convergence modes and when each is sufficient.
Session 12 — April 18, 2026
Chapter: Billingsley, Probability and Measure, Chapter 27 — Conditional Expectation
Presenter: Khue
Summary: Defined conditional expectation as a Radon–Nikodym derivative. Worked through examples showing how this generalizes the elementary definition.
Session 11 — April 11, 2026
Chapter: Billingsley, Probability and Measure, Chapter 26 — Product Measures and Fubini’s Theorem
Presenter: Maria
Summary: Constructed product measures and proved Fubini–Tonelli. Applied it to compute expectations of functions of independent random variables.
Session 10 — April 4, 2026
Chapter: Billingsley, Probability and Measure, Chapters 24–25 — Lp Spaces
Presenter: Alex
Summary: Covered Hölder’s and Minkowski’s inequalities, completeness of Lp spaces. Discussed the role of L2 in least-squares estimation.
Session 9 — March 28, 2026
Chapter: Billingsley, Probability and Measure, Chapter 22 — The Radon–Nikodym Theorem
Presenter: Sam
Summary: Proved the Radon–Nikodym theorem for sigma-finite measures. Connected it to densities and likelihood ratios.
Session 8 — March 21, 2026
Chapter: Billingsley, Probability and Measure, Chapter 20 — Random Variables and Expected Values
Presenter: Khue
Summary: Formally defined random variables as measurable functions. Built up the Lebesgue integral for expected values, connecting to the abstract integration theory.
Session 7 — March 14, 2026
Chapter: Billingsley, Probability and Measure, Chapter 17 — Dominated and Monotone Convergence
Presenter: Jordan
Summary: Proved DCT and MCT. Discussed Fatou’s lemma as the bridge between them and worked through counterexamples when hypotheses fail.
Session 6 — March 7, 2026
Chapter: Billingsley, Probability and Measure, Chapters 15–16 — The Integral
Presenter: Maria
Summary: Constructed the Lebesgue integral via simple functions. Compared with the Riemann integral and showed examples of Lebesgue-integrable functions that aren’t Riemann-integrable.
Session 5 — February 28, 2026
Chapter: Billingsley, Probability and Measure, Chapter 13 — Measurable Functions
Presenter: Alex
Summary: Defined measurable functions and proved closure under limits. Connected to the definition of random variables.
Session 4 — February 21, 2026
Chapter: Billingsley, Probability and Measure, Chapters 10–12 — Lebesgue Measure on the Real Line
Presenter: Sam
Summary: Constructed Lebesgue measure via outer measure and the Carathéodory extension. Discussed non-measurable sets (Vitali) and the axiom of choice.
Session 3 — February 14, 2026
Chapter: Billingsley, Probability and Measure, Chapter 4 — Borel Sets and Lebesgue Measure
Presenter: Khue
Summary: Introduced Borel sigma-algebras and their role in probability. Discussed the construction of probability measures on the real line via distribution functions.
Session 2 — February 7, 2026
Chapter: Billingsley, Probability and Measure, Chapters 2–3 — Probability Measures and Existence
Presenter: Jordan
Summary: Defined probability measures on sigma-algebras and proved basic properties (continuity, inclusion-exclusion). Discussed the Carathéodory extension theorem.
Session 1 — January 31, 2026
Chapter: Billingsley, Probability and Measure, Chapter 1 — Borel’s Normal Number Theorem
Presenter: Khue
Summary: Kicked off with Billingsley’s motivating example — proving that almost all numbers are normal using measure-theoretic tools. Set the stage for why we need this machinery.