From Lecture Notes to Lean: Formalizing a Textbook on Probability Theory

Abstract

As large language models become increasingly capable of generating mathematical arguments, mathematics is likely to face not a scarcity of proofs but an abundance of plausible ones. In such an environment, verification, exposition, and incorporation into reusable mathematical infrastructure become central tasks. We report on an ongoing Lean formalization of"Measure-Theoretic Probability: With Applications to Statistics, Finance, and Engineering", a fourteen-chapter upper-level undergraduate textbook covering topics from Riemann--Stieltjes integration to martingales and limit theorems. The project produces a machine-checked companion to the textbook and contributes reusable infrastructure for future formalizations involving probability theory. A Lean formalization provides computer-checked statements and proofs, makes hypotheses explicit, and allows readers to inspect the precise logical content of textbook results. A central challenge is to bridge textbook-facing statements with Mathlib's more general measure-theoretic interfaces. We reuse Mathlib results when possible and introduce reviewable interface lemmas when the textbook formulation and library abstraction differ. The project illustrates how formalized textbooks can support teaching, clarify mathematical assumptions, and help build the formal foundations needed for reliable AI-assisted mathematics.