Synopses & Reviews
This graduate-level textbook and monograph defines the functions of a real variable through consistent use of the Daniell scheme, offering a rare and useful alternative to customary approaches. The treatment can be understood by any reader with a solid background in advanced calculus, and it features many problems with hints and answers. "The exposition is fresh and sophisticated," declared Sci-Tech Book News,
"and will engage the interest of accomplished mathematicians."
Part one is devoted to the integral, moving from the Reimann integral and step functions to a general theory, and obtaining the "classical" Lebesgue integral in n space. Part two constructs the Lebesgue-Stieltjes integral through the Daniell scheme using the Reimann-Stieltjes integral as the elementary integral. Part three develops theory of measure with the general Daniell scheme, and the final part is devoted to the theory of the derivative.
This treatment examines the general theory of the integral, Lebesque integral in n space, the Riemann-Stieltjes integral, and more. "The exposition is fresh and sophisticated, and will engage the interest of accomplished mathematicians." — Sci-Tech Book News. 1966 edition.
Table of Contents
Introduction Part 1 The Integral 1. The Riemann Integral and Step Functions 2. General Theory of the Integral 3. The Lebesgue Integral in n-Space Part 2 The Stieltjes Integral 4. The Riemann-Stieltjes Integral 5. The Lebesgue-Stieltjes Integral 6. Measurable Sets and General Measure Theory 7. Construtive Measure Theory 8. Axiomatic Measure Theory 9. Measure and Set Functions 10. The Derivative of a Set Function Bibliography Index