## Analysis I Prelim topics

## Topics

Solid grounding in undergraduate analysis at the level of Rudin’s analysis book.

Measures (outer measure on R, measurable spaces and functions, properties of measures, Lebesgue measure, convergence of measurable functions, Egorov's and Luzin's theorems)

- Integration (integration with respect to a measure, monotone convergence theorem, limits of integrals and integrals of limits, dominated convergence theorem, Riemann and Lebesgue integrals)
- Differentiation (Hardy-Littlewood maximal function and inequality, Lebesgue differentiation theorem)
- Product measures (products of measure spaces, iterated integrals, Tonelli's and Fubini's theorems, Lebesgue measure/integration on R^n)
- Banach spaces (normed vector spaces, bounded linear maps, linear functionals, Hahn-Banach theorems, consequences of Baire's theorem (open mapping theorem, inverse mapping theorem, closed graph theorem, and principle of uniform boundedness))
L^p spaces (Holder's and Minkowski's inequalities, L^p as a Banach space, duality).

## Main References

- Walter Rudin, “Principles of Mathematical Analysis”
- S. Axler, ``Measure, integration & real analysis".
- H. Royden, P. Fitzpatrick, ``Real analysis" (4th edition).