Introduction to Measure and Integration Theory

Abstract

These notes provide a rigorous and accessible introduction to measure and integration theory, with emphasis on the conceptual transition from the Riemann integral to the Lebesgue integral and the role played by limiting processes in modern analysis. The manuscript develops the basic theory of measurable sets, measurable functions, measures on $\sigma$-algebras, Lebesgue integration, convergence theorems, and $L^p$ spaces. Particular attention is devoted to the interaction between integration and convergence, as well as to the limitations of the Riemann integral that motivate the development of measure theory. The exposition seeks to balance mathematical rigor with pedagogical clarity through detailed proofs, examples, exercises, and supplementary projects. These notes are intended primarily for undergraduate students in mathematics and related areas encountering measure theory for the first time, although they may also serve as a reference for introductory graduate courses in analysis.