--- title: Fubini’s Theorem description: This fundamental result in measure theory establishes conditions under which the order of integration can be switched in multiple integrals. Learners will understand how to compute iterated integrals and apply this theorem to multi-dimensional spaces. category: mathematics subcategory: measure-theory difficulty: beginner, intermediate, advanced url: /subject/fubinis-theorem --- # Fubini’s Theorem This fundamental result in measure theory establishes conditions under which the order of integration can be switched in multiple integrals. Learners will understand how to compute iterated integrals and apply this theorem to multi-dimensional spaces. ## Where to start Start with section 1.7 of Terence Tao's free An Introduction to Measure Theory, which builds product measures and proves the Tonelli and Fubini theorems with the sigma-finiteness hypotheses made explicit. If you only use one resource, make it Gerald B. Folland's Real Analysis, a paid graduate textbook whose section 2.5 treats product measures and Fubini-Tonelli. Adam Kashlak's recorded lecture, Probability and Measure, Lecture 8, works through the full proofs. ## Available Resources 2 Books • 1 Courses • 3 Websites ## Websites ### 1. Math Stack Exchange: Fubini's Theorem Discussions Math Stack Exchange's question archive for Fubini's theorem, where answered questions work through when iterated integrals may be swapped, counterexamples where the theorem's hypotheses fail, and the related Tonelli theorem. Useful for checking specific exercises and sharpening understanding of the integrability conditions. **Difficulty:** Beginner | **Price:** Free **Link:** https://math.stackexchange.com/questions/tagged/fubini%27s-theorem **Tags:** fubini-theorem, iterated-integrals, measure-theory, real-analysis, tonelli-theorem ### 2. Terence Tao's Notes on Measure Theory Terence Tao's blog posts and lecture notes on measure theory, including his graduate real analysis notes on Lebesgue integration, product measures and the Fubini–Tonelli theorem. Working through them gives a rigorous account of when integrals over product spaces can be computed iteratively. **Difficulty:** Beginner | **Price:** Free **Link:** https://terrytao.wordpress.com/tag/measure-theory/ **Tags:** measure-theory, lebesgue-integration, product-measures, fubini-theorem, real-analysis ### 3. Wolfram MathWorld **Author:** Eric W. Weisstein 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Courses ### 1. Real Analysis (MIT 18.100A) **Author:** Casey Rodriguez 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. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ **Tags:** real-analysis, proofs, sequences-and-series, uniform-convergence, riemann-integral ## Youtubes ### 1. Probability and Measure, Lecture 8: Product Measure and Fubini-Tonelli **Author:** Adam Kashlak This is a real university lecture that proves the theorem rather than sketching it. Recorded graduate lecture, roughly two and a quarter hours, proving existence and uniqueness of the product measure, then the Fubini and Tonelli theorems, then existence and uniqueness of probability measures on infinite product spaces. Typed notes linked. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/watch?v=s4XU4ZHTyAM **Tags:** measure-theory, product-measure, fubini-tonelli, probability-theory, lecture ## Books ### 1. Real Analysis: Modern Techniques and Their Applications (2nd ed.) **Author:** Gerald B. Folland The reference every graduate course cites for the abstract Fubini-Tonelli statement. Standard graduate reference on measure, integration, and functional analysis. Chapter 2 develops the abstract integral, with section 2.5 treating product measures and the Fubini-Tonelli theorem for sigma-finite spaces, including the completion subtleties on R^n. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0471317160?tag=edmonddante07-20 **Tags:** real-analysis, measure-theory, lebesgue-integration, fubini-tonelli, functional-analysis ### 2. An Introduction to Measure Theory (free PDF) **Author:** Terence Tao Graduate text covering Lebesgue measure, abstract measure spaces, and the convergence theorems. Section 1.7 builds product measures from outer measures and pre-measures, then proves the Tonelli and Fubini theorems with the sigma-finiteness hypotheses made explicit. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://terrytao.wordpress.com/wp-content/uploads/2012/12/gsm-126-tao5-measure-book.pdf **Tags:** measure-theory, lebesgue-integration, product-measure, fubini-tonelli, real-analysis --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/fubinis-theorem