## 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%