Measure Theory and Modern Analysis

This monograph provides a rigorous yet accessible introduction to measure theory and modern analysis. Starting from outer measures and Carathéodory’s extension theorem, it develops the foundations of Lebesgue measure, measurable functions, and the Lebesgue integral.

The text covers the main convergence theorems, LpL^pLp spaces, product measures, and the Fubini–Tonelli theorems, with a clear emphasis on conceptual structure and proofs. Further chapters introduce induced measures and change-of-variables formulas, probability from a measure-theoretic viewpoint, Fourier analysis, concentration of measure, and the law of large numbers and the central limit theorem.

Historical notes are included throughout to place the results in context, and exercises at the end of each chapter reinforce key ideas. The book is intended for advanced undergraduate and graduate students in mathematics, as well as researchers seeking a concise and coherent reference on the measure-theoretic foundations of modern analysis and probability.