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Riesz theorem. Almost uniform convergence. Section - III. Shortcomings of Riemann Integral, Lebesgue Integral of a bounded function over a set of finite measure. Chapter 3, measure theory, is the essence of this book. Here an elementary, but rigorous, treatment of Lebesgue measure, as a natural extension of "length of an Schaum's outline of theory and problems of real variables; Lebesgue measure and integration with applications to Fourier series. Chapter 3, measure theory, is the essence of this book. Here an elementary, but rigorous, treatment of Lebesgue measure, as a natural extension of "length of an We have constructed the Lebesgue measure µ which assigns a d-dimensional volume to a large class of subset of Rd, and we have explored the basic properties of a line, and the measure Lebesgue measure. In keeping with the role of the book as a first text on the subject, the proofs are set out in considerable detail.The aim of this book is to cover a basic course of. Lebesgue measure and integration which flows in a natural' moti- vated and simple way. The authors have