Lebesgue Integral: An Elementary Approach

Using the Daniell-Riesz approach, this text presents the Lebesgue integral at a level accessible to an audience familiar only with limits, derivatives and series. Employing such minimal prerequisites allows for greatly increased curricular flexibility for course instructors, as well as providing undergraduates with a gateway to the powerful modern mathematics of functions at a very early stage. The book's topics include: the definition and properties of the Lebesgue integral; Banach and Hilbert spaces; integration with respect to Borel measures, along with their associated L2(μ) spaces; bounded linear operators; and the spectral theorem. The text also describes several applications of the theory, such as Fourier series, quantum mechanics, and probability.