This book has two main themes： the Baire cate8ory theorem as a method for proving existence， and the \"duality\" between measure and category. The category method is illustrated by a variety of typical applications， and the analogy between measure and category is explored in all of its ramifications. To this end， the elements of metric topology are reviewed and the principal properties of Lebesgue measure are derived. It turns out that Lebesgue integration is not essential for present purposes-the Riemann integral is sufficient. Concepts of general measure theory and topology are introduced， but not just for the sake of 8cncrality， Needless to say， the term \"category\" refers always to Bairc category； it has nothing to do with the term as it is used in homological algebra.
　　The book is a revised and expanded version of notes originally prepared for a course of lectures given at Haverford College during the spring of 1957 under the auspices of the William Pyle Philips Fund. These， in turn， were based on the Earle Raymond Hedrick Lectures presented at the Summer Meeting of the Mathematical Association of America at Seattle， Washington， in August， 1956.

　　This book has two main themes： the Baire cate8ory theorem as a method for proving existence， and the \"duality\" between measure and category. The category method is illustrated by a variety of typical applications， and the analogy between measure and category is explored in all of its ramifications. To this end， the elements of metric topology are reviewed and the principal properties of Lebesgue measure are derived. It turns out that Lebesgue integration is not essential for present purposes-the Riemann integral is sufficient. Concepts of general measure theory and topology are introduced， but not just for the sake of 8cncrality， Needless to say， the term \"category\" refers always to Bairc category； it has nothing to do with the term as it is used in homological algebra.
　　A knowledge of calculus is presupposed， and some familiarity with the algebra of sets. The questions discussed are ones that lend themselves naturally to set-theoretical formulation， The book is intended as an introduction to this kind， of analysis. It could be used to supplement a standard course in real analysis， as the basis for a seminar， or for independent study. It is primarily expository， but a few refinements of known results are included， notably Theorem 15.6 and Proposition 20.4. The references are not intended to be complete. Frequently a secondary source is cited where additional references may be found.
　　The book is a revised and expanded version of notes originally prepared for a course of lectures given at Haverford College during the spring of 1957 under the auspices of the William Pyle Philips Fund. These， in turn， were based on the Earle Raymond Hedrick Lectures presented at the Summer Meeting of the Mathematical Association of America at Seattle， Washington， in August， 1956.