This textbook is designed for a year-long course in real analysis taken by beginning graduate and advanced undergraduate students in mathematics and other areas such as statistics, engineering, and economics. Written by one of the leading scholars in the field, it elegantly explores the core concepts in real analysis and introduces new, accessible methods for both students and instructors. The first half of the book develops both Lebesgue measure and, with essentially no additional work for the student, general Borel measures for the real line. Notation indicates when a result holds only for Lebesgue measure. Differentiation and absolute continuity are presented using a local maximal function, resulting in an exposition that is both simpler and more general than the traditional approach. The second half deals with general measures and functional analysis, including Hilbert spaces, Fourier series, and the Riesz representation theorem for positive linear functionals on continuous functions with compact support. To correctly discuss weak limits of measures, one needs the notion of a topological space rather than just a metric space, so general topology is introduced in terms of a base of neighborhoods at a point. The development of results then proceeds in parallel with results for metric spaces, where the base is generated by balls centered at a point. The text concludes with appendices on covering theorems for higher dimensions and a short introduction to nonstandard analysis including important applications to probability theory and mathematical economics.
This textbook is designed for a year-long course in real analysis taken by beginning graduate and advanced undergraduate students in mathematics and other areas such as statistics, engineering, and economics. Written by one of the leading scholars in the field, it elegantly explores the core concepts in real analysis and introduces new, accessible methods for both students and instructors. The first half of the book develops both Lebesgue measure and, with essentially no additional work for the student, general Borel measures for the real line. Notation indicates when a result holds only for Lebesgue measure. Differentiation and absolute continuity are presented using a local maximal function, resulting in an exposition that is both simpler and more general than the traditional approach. The second half deals with general measures and functional analysis, including Hilbert spaces, Fourier series, and the Riesz representation theorem for positive linear functionals on continuous functions with compact support. To correctly discuss weak limits of measures, one needs the notion of a topological space rather than just a metric space, so general topology is introduced in terms of a base of neighborhoods at a point. The development of results then proceeds in parallel with results for metric spaces, where the base is generated by balls centered at a point. The text concludes with appendices on covering theorems for higher dimensions and a short introduction to nonstandard analysis including important applications to probability theory and mathematical economics.
The aim of this book is to make Robinson's discovery, and some of the subsequent research, available to students with a background in undergraduate mathematics. In its various forms, the manuscript was used by the second author in several graduate courses at the University of Illinois at Urbana-Champaign. The first chapter and parts of the rest of the book can be used in an advanced undergraduate course. Research mathematicians who want a quick introduction to nonstandard analysis will also find it useful. The main addition of this book to the contributions of previous textbooks on nonstandard analysis (12,37,42,46) is the first chapter, which eases the reader into the subject with an elementary model suitable for the calculus, and the fourth chapter on measure theory in nonstandard models.
A NEW YORKER BEST BOOK OF THE YEAR • A revolutionary new history that reveals how climate change has dramatically shaped the development—and demise—of civilizations across time *The ebook edition now includes endnotes. Anyone who purchased the book previously can re-download this updated edition and access the notes.* Global warming is one of the greatest dangers mankind faces today. Even as temperatures increase, sea levels rise, and natural disasters escalate, our current environmental crisis feels difficult to predict and understand. But climate change and its effects on us are not new. In a bold narrative that spans centuries and continents, Peter Frankopan argues that nature has always played a fundamental role in the writing of history. From the fall of the Moche civilization in South America that came about because of the cyclical pressures of El Niño to volcanic eruptions in Iceland that affected Egypt and helped bring the Ottoman empire to its knees, climate change and its influences have always been with us. Frankopan explains how the Vikings emerged thanks to catastrophic crop failure, why the roots of regime change in eleventh-century Baghdad lay in the collapse of cotton prices resulting from unusual climate patterns, and why the western expansion of the frontiers in North America was directly affected by solar flare activity in the eighteenth century. Again and again, Frankopan shows that when past empires have failed to act sustainably, they have been met with catastrophe. Blending brilliant historical writing and cutting-edge scientific research, The Earth Transformed will radically reframe the way we look at the world and our future.
A small town on the Mexican border draws a colorful array of characters in search of romance, sanctuary, fame, or fortune, including Mackie Larkin, a suburban mother desperate for money; her ex-husband Keith, a man with a host of get-rich schemes; their impulsive teenage daughter, Lianne; charismatic strip-club owner Buck Samsonov; and Nicholas Loeb, a struggling mystery writer. Reprint.
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