Volume and Integral

"This introduction to the theory of the Lebesgue Integral is primarily intended for third-year Honours students. A consistently geometrical approach has been chosen in order to show the simple underlying ideas. First, the volume of an n-dimensional set is defined as it s Lebesgue Measure. The integral of a non-negative function of n real variables is then the volume of its ordinate set; the extension to general functions follows easily. The old notion of the Riemann Integral is similarly developed; its relation to the Lebesgue Integral and the striking advantages of the latter are clearly set out." The main object of this book is to provide an introduction to the theory of the so-called absolute integral. It is not an introduction to the "calculus": it is assumed that the reader is familiar with this. But the book should give the student a deeper understanding of the ideas underlying the calculus. It is also hoped that they will appreciate the aesthetic side of a purely mathematical theory, quite apart from its practical implications. Such an appreciation is quite as essential as technical skill.