Measure Theory and Integration By and For the Learner

Abstract

Measure Theory and Integration is exposed with the clear aim to help beginning learners to perfectly master its essence. In opposition of a delivery of the contents in an academic and vertical course, the knowledge is broken into exercises which are left to the learners for solutions. Hints are present at any corner to help readers to achieve the solutions. In that way, the knowledge is constructed by the readers by summarizing the results of one or a group of exercises. Each chapter is organized into Summary documents which contain the knowledge, Discovery documents which give the learner the opportunity to extract the knowledge himself through exercises and into Solution Documents which offer detailed answers for the exercises. Exceptionally, a few number of results (A key lemma related the justification of definition of the integral of a non-negative function, the Caratheodory's theorem and the Lebesgue-Stieljes measure on $\mathbb{R}^d$) are presented in appendix documents and given for reading in small groups. The full theory is presented in the described way. We highly expect that any student who goes through the materials, alone or in a small group or under the supervision of an assistant will gain a very solid knowledge in the subject and by the way ensure a sound foundation for studying disciplines such as Probability Theory, Statistics, Functional Analysis, etc. The materials have been successfully used as such in normal real analysis classes several times.