### Course Meeting Times

Lectures: 3 lectures / week, 1 hours / lecture

### Prerequisites

*18.100 Analysis I*; *18.06 Linear Algebra*, *18.700 Linear Algebra*, or *18.701 Algebra I*

### Textbooks

The primary text book is Adams and Guillemin, but we will also refer to Stein and Shakarchi later in the semester.

Adams, Malcolm Ritchie, and Victor Guillemin. *Measure Theory and Probability*. Birkhäuse, 1996. ISBN: 9780817638849. [Preview with Google Books]

Stein, Elias M., and Rami Shakarchi. *Fourier Analysis: An Introduction*. Princeton University Press, 2003. ISBN: 9780691113845.

Another useful book for reference is:

Körner, T. W. *Fourier Analysis*. Cambridge University Press, 1988. ISBN: 9780521251204.

This book is a series of vignettes that make entertaining reading in small doses. We will not be using it, but it gives an idea of the range of applications of Fourier analysis.

### Course Description

This course continues the content covered in *18.100 Analysis I*. Roughly half of the subject is devoted to the theory of the Lebesgue integral with applications to probability, and the other half to Fourier series and Fourier integrals.

The task in the first half of the course is to introduce Lebesgue measure and establish properties of the Lebesgue integral. Our textbook (Adams and Guillemin) introduces Lebesgue measure using motivation and examples from probability theory. After we have developed probability theory on Bernoulli sequences, using a corresponding with Lebesgue measure on the unit interval, we will discuss the Lebesgue integral and some Fourier analysis. Then we will use some Fourier analysis to prove more theorems in probability. By the end of the semester we will have all the tools to discuss the continuum limit of a (suitably scaled) random walk, namely Brownian motion.

One of the main goals this course is to establish rules for the limiting behavior of functions so that we can deal with functions with as much confidence as we do real or complex numbers. An equally important motivation (that will only become clear in the second half) is that the systematic study of Fourier series requires the Lebesgue integral. The square mean convergence of Fourier series and Parseval’s formula cannot be stated accurately in proper generality without the Lebesgue integral and Lebesgue integrable functions.

### Assignments and Exams

There will be 11 problem sets, due at the beginning of class on the due dates. Late homework will be accepted only if it is turned in within one week of the due date. There is no penalty for the first late homework assignment, but scores of all subsequent late papers will be multiplied by 1/2. Collaboration on problem sets is encouraged, but read each problem carefully and make an attempt to solve it by yourself. You must write up all homework problems by yourself.

There will be one in-class midterm test and a final exam.

### Grading

ACTIVITIES | PERCENTAGES |
---|---|

Problem sets | 30% |

Midterm exam | 30% |

Final exam | 40% |

### Calendar

LEC # | TOPICS | KEY DATES |
---|---|---|

1 | Coin Tossing, Law of Large Numbers, Rademacher Functions | Problem Set 1 Due |

2 | Measure Theory, Random Models | Problem Set 2 Due |

3 | Measurable Functions, Lebesgue Integral | Problem Set 3 Due |

4 | Convergence Theorems, Riemann Integrability | Problem Set 4 Due |

5 | Fubini’s Theorem, Independent Random Variables | Problem Set 5 Due |

6 | Lebesgue Spaces, Inner Products | Problem Set 6 Due |

7 | Hilbert Space, Midterm Review | Midterm Test |

8 | Fourier Series and their Convergence | Problem Set 7 Due |

9 | Applications of Fourier Series | Problem Set 8 Due |

10 | Fourier Integrals | Problem Set 9 Due |

11 | Fourier Integrals of Measures, Central Limit Theorem | Problem Set 10 Due |

12 | Brownian Motion | Problem Set 11 Due |

13 | Brownian Motion Concluded, Review for Final Exam | Final Exam |