Measure Theory and Lebesgue Integration

The Lebesgue integral is a generalization of the Riemann integral which extends the collection of functions which are integrable. Lebesgue integration differs from Riemann integration in the way the approximations to the integral are taken. Riemann approximations use step functions which have a constant value on any given interval of the domain corres­ponding to some partition. Lebesgue approximations use what are called simple functions which, like the step functions, take on only a finite number of values. However, these values are not necessarily taken on by the function on intervals of the domain, but rather on arbitrary subsets of the domain. The integration of simple functions under the most general circumstances possible necessitates a generalization of our notion of length of a set when the set is more complicated than a simple interval. We define the Iebesgue measure "m" of a set E Є M, where M is some collection of sets of real numbers, to be a certain set function which assigns to Ea nonnegative extended real number 'mE '. This report consists of the solutions of exercises found in 'Real Analysis", by H. L. Hoyden. Quotations from the book are all accompanied by the title "Definition" or "Theorem". The exercises are all entitled "Proposition" and all proofs in this report are my own. All theorems are quoted without proof The theorems and definitions occur as they are needed throughout the paper1 but some of the most basic definitions and theorems are lumped together in section II. It is assumed in this paper that the reader is familiar with the basic concepts of advanced calculus and set theory,