## Introduction to IntegrationIntroduction to integration provides a unified account of integration theory, giving a practical guide to the Lebesgue integral and its uses, with a wealth of illustrative examples and exercises. The book begins with a simplified Lebesgue-style integral (in lieu of the more traditionalRiemann integral), intended for a first course in integration. This suffices for elementary applications, and serves as an introduction to the core of the book. The final chapters present selected applications, mostly drawn from Fourier analysis. The emphasis throughout is on integrable functionsrather than on measure. The book is designed primarily as an undergraduate or introductory graduate textbook. It is similar in style and level to Priestley's Introduction to complex analysis, for which it provides a companion volume, and is aimed at both pure and applied mathematicians.Prerequisites are the rudiments of integral calculus and a first course in real analysis. |

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### Contents

Setting the scene | 1 |

Preliminaries | 8 |

Intervals and step functions | 21 |

Integrals of step functions | 29 |

Continuous functions on compact intervals | 34 |

Techniques of integration I | 44 |

Approximations | 56 |

Uniform convergence and power series | 67 |

Measurable functions | 160 |

Measurable sets | 166 |

The character of integrable functions | 172 |

Integration vs differentiation | 177 |

Integrable functions on Rfc | 189 |

Fubinis Theorem and Tonellis Theorem | 190 |

Transformations of Rk | 209 |

The spaces L1 L2 and L? | 210 |

Building foundations | 78 |

Null sets | 87 |

Linc functions | 93 |

The class L of integrable functions | 102 |

Nonintegrable functions | 110 |

MCT and DCT | 117 |

Recognizing integrable functions I | 125 |

Techniques of integration II | 132 |

Sums and integrals | 137 |

Recognizing integrable functions II | 143 |

The Continuous DCT | 148 |

Differentiation of integrals | 152 |

pointwise convergence | 221 |

convergence reassessed | 236 |

orthogonal sequences | 247 |

L2spaces as Hilbert spaces | 258 |

Fourier transforms | 264 |

Integration in probability theory | 279 |

historical remarks | 287 |

reference | 291 |

295 | |

297 | |

299 | |

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### Common terms and phrases

antiderivative apply the MCT approximation assume Borel set bounded interval calculation Cauchy Chapter compact interval Comparison Theorem complete consider constant Continuous DCT continuous function convergence theorems converges uniformly countable Deduce defined definition derivative disjoint elementary endpoints evaluate Exercise example exists exponential finite fixed fn(x formula Fourier series Fourier transform Fubini's Theorem function g G Lr given Hence implies Indefinite Integral inequality infinite inner product space integrable functions integration theory Lebesgue integral Lemma Let f(x limit Linc-sequence linear logx Lstep measurable function measurable sets non-negative normed space notation null set open intervals orthogonal partial sums pn(x Proof properties Prove random variable real numbers rectangle repeated integrals result sinx sn(x step function subintervals subset Technical Theorem Tonelli's Theorem trapezium rule uniform convergence union write zero