This course is designed to introduce students to the theory of Lebesgue measure and integration. We'll begin by examining some defects of the Riemann integral and then develop Lebesgue's ideas of how to circumvent these defects. Although Lebesgue did not complete the task, his theory forms the basis for the modern theory of integration and an inherent measure theory.

This "measure theory" is not just probabilistic in nature, it is the foundation of modern probability theory as we now know it. Often during the semester, I'll use probability as motivation for definitions and proofs.

Our course will follow the historical line of thought which, although not the most direct route, reveals much about how the theory of integration is closely tied to the geometry of functions and the foundations of probability. Topics covered include:

• Definition of the Lebesgue Integral and Basic Results
• Measurable Sets and Functions
• Convergece Theorems
• Fourier Representation of Functions
• Basic L^p Space Theory

But for goals, I have two and only two; to learn a little bit and to have some fun doing just that!

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