Real Analysis

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This textbook is tailored for a year-long real analysis course aimed at beginning graduate and advanced undergraduate students in mathematics, statistics, engineering, and economics. Authored by a leading scholar, it explores core real analysis concepts and presents new, accessible methods for students and instructors. The first half focuses on Lebesgue measure and general Borel measures for the real line, clearly indicating when results apply solely to Lebesgue measure. Differentiation and absolute continuity are introduced using a local maximal function, offering a simpler and more general exposition than traditional methods. The second half covers general measures and functional analysis, including topics like Hilbert spaces, Fourier series, and the Riesz representation theorem for positive linear functionals on continuous functions with compact support. To address weak limits of measures, the text introduces general topology through neighborhoods at a point, progressing alongside results for metric spaces. The book concludes with appendices on covering theorems for higher dimensions and a brief introduction to nonstandard analysis, highlighting its significant applications in probability theory and mathematical economics.