John Möller: The Lebesgue Integral

Abstract: The aim of this thesis is to develop the Lebesgue integral and to study its fundamental convergence properties. Beginning with measure theory and measurable functions, the Lebesgue integral is constructed and its basic properties are established. The Monotone Convergence Theorem, Fatou’s Lemma, and the Dominated Convergence Theorem are proved and used to explore key structural features of the space L1. These results are then applied to integrals with respect to measures defined by densities and to the formulation of probability theory in a measure-theoretic framework. As a final application, the Central Limit Theorem is proved using methods from Fourier analysis.