An Introduction to Measure Theory
by Terence Tao · Terence Tao
Graduate textbook (AMS Graduate Studies in Mathematics 126) that builds Lebesgue measure and integration concretely before abstract measure spaces, the Carathéodory extension theorem, Fubini's theorem, and differentiation theorems. Prepares readers for graduate real analysis and rigorous probability theory.
This link may earn us a small commission at no extra cost to you. Affiliate disclosure
More resources on Measure Theory
An Introduction to Measure Theory
Graduate textbook by Terence Tao building Lebesgue measure and integration on the real line and Euclidean space, then abstract measure spaces, product measures and Fubini, and differentiation theorems. Readers learn to prove the core convergence theorems and work rigorously with measurable functions.
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Fremlin's Measure Theory
D. H. Fremlin's five-volume treatise, free in draft form from the author's Essex page. Works from the construction of Lebesgue measure and integration through measure algebras, topological measure spaces and set-theoretic measure theory, with complete proofs for reference-level study.
Sheldon Axler's Book Site
Official site for Sheldon Axler's Measure, Integration & Real Analysis (Springer Graduate Texts in Mathematics 282), with the complete book free as an open-access PDF. It develops Lebesgue measure, abstract integration, Lp spaces, Hilbert spaces, and Fourier analysis from first principles.
Real Analysis (MIT 18.100A)
Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs.
Projection Theory (MIT 18.156)
Graduate course on how sets behave under orthogonal projections, from first questions to recent research, with applications to harmonic analysis, analytic number theory, additive combinatorics and homogeneous dynamics. Includes 24 lecture videos, lecture notes and problem sets for readers comfortable with graduate analysis.