Topics:

• \(\sigma\)-algebra and measures: definition of measures, outer measures, Caratheodory theorem, extension of measures and construction of Lebesgue measure, Borel measures on \(\mathbb{R}\) and \(\mathbb{R}^n\).
• Integration and \(L^p\) space: measurable functions, definition of integrals, \(L^1\) space, modes of convergence, product measure and Fubini-Tonelli Theorem. \(L^p\) space and inequalities in \(L^p\) spaces.
• Signed measure, Lebesgue-Radon-Nykodim theorem, differentiation and absolute continuity, Hardy-LIttlewood maximal function.
• Some topological prerequisites: topology, neighborhood, countability and separation axioms, continuity.
• \(L^p\) space as Banach spaces: introduction to Banach spaces, linear functionals, dual of \(L^p\) spaces.
• Continuous functions and \(L^p\) functions: LCH spaces, Stone-Weierstrass theorem, Radon measures, the Riesz-Markov Theorem.
• Supplementary materials (optional): Haar measure, Hausdorff measure and self similar measures.

Textbook reference:

Folland, Real Analysis: Modern Techniques and Their Applications. Wiley, 1999.

We will cover the following sections in Folland (although the presentation may differ and we will not cover them in the same order): (1.1) - (1.5), (2.1) - (2.6), (3.1) - (3.4), part of (3.5), (5.1) - (5.2), (6.1) - (6.2), part of (7.1) - (7.3). Optional: (11.1) - (11.3).

References

• Royden, Fitzpatrck, Real Analysis. Prentice Hall, 2010.
  • A complete reference including set theoretic axioms and real number theory.
  • Contains more topics than we cover, good for supplementary reading.
• Halmos, Measure Theory. Springer, 2013.
  • Detailed, slow paced exposition of measure theory.
• Tao, An Introduction to Measure Theory. American Mathematical Society. 2011.
  • Starts with \(\mathbb{R}^n\).
  • Many exercises.
  • A shorter book based on a quarter course.

Evaluation

• Homework: 30%
  
• Midterm Test: 30%
• Final Exam: 40%