An Introduction to the Gauge Integral
by Eric Schechter · Vanderbilt University
Short expository page on why the Riemann integral fails under pointwise limits, what the Lebesgue integral fixes and at what prerequisite cost, and how the gauge integral reaches comparable strength from a definition barely different from Riemann's. Includes a containment diagram.
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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.
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.
Lebesgue's Theory of Integration: Its Origins and Development
Traces how nineteenth-century breakdowns of the Riemann integral, including nowhere-dense sets of positive content, term-by-term integration of series, and the failure of the fundamental theorem, forced the concepts Lebesgue assembled in 1902. Prize-winning history of the integral.
A Garden of Integrals
Compares eight integrals in one volume: Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Stieltjes, Henstock-Kurzweil, Wiener, and Feynman. Shows what each can integrate, how the classes nest, and which convergence and fundamental-theorem results survive in each framework.
A Modern Theory of Integration
The single most integration-native book in the field: it is about repairing the integral itself. It answers the exact question the topic poses — what is wrong with Riemann, what does Lebesgue fix, and what does a third framework buy — and reverses the usual dependency by getting measure theory out of the integral. Develops the Henstock-Kurzweil (gauge) integral from a Riemann-style definition, then derives measure theory as a consequence rather than a prerequisite. Covers the fundamental theorems of calculus, the Saks-Henstock lemma, absolute integrability, convergence theorems, and modes of convergence.