An Analytical Study of Lebesgue Integration and Its Applications

Abstract

This paper presents an overview of the theory of Lebesgue Integration, which is considered one of the fundamental concepts in modern mathematical analysis. The study introduces the basic notions of measurable sets and measurable functions and discusses the construction and main properties of the Lebesgue integral. In addition, several fundamental convergence theorems, including the monotone convergence theorem, Fatou’s lemma, and the dominated convergence theorem, are presented.

Furthermore, the paper explores a range of applications of the Lebesgue integral in probability theory, statistics, functional analysis, Fourier analysis, and partial differential equations. These applications are discussed from an analytical perspective, highlighting the role of Lebesgue integration in handling limits, convergence, and irregular functions. The results demonstrate the importance of this theory as a powerful and flexible tool in modern mathematical research.