# Real Analysis Exams¶

These exams were taken by graduate students in mathematics at the University of Hawaii at Manoa.

## Appendices and Indices¶

**Topics.**Expand to see a list of some topics covered on the exams.

**Functions**

- elementary topology of \(\mathbb R^n\)
- continuous functions
- uniformly continuous functions
- absolutely continuous functions
- differentiability, implicit function theorem
- functions of bounded variation, rectifiable curves
- Borel functions
- measurable functions
- simple functions
- relations between these classes, Lusin’s theorem

**Integration**

- Lebesgue integral
- Fatou’s lemma
- monotone and dominated convergence theorems
- applications to moving limits through integrals
- differentiation under an integral sign
- product measures, Fubini-Tonelli theorem
- definition and completeness of \(L^p\) spaces
- Riesz representation theorem
- Lebesgue differentiation theorem

**Convergence of functions**

- pointwise convergence
- supremum norm, uniform convergence
- convergence in measure
- convergence in \(L^p\) spaces
- relations between these notions, Egorov’s theorem

**Inequalities**

- Chebyshev
- Cauchy-Schwarz
- Jensen
- Hölder
- Minkowski (sum and integral forms)

**Density**

- Weierstrass theorem, density of polynomials in \(L^p\) spaces
- convolution with approximate identities
- density of smooth functions in \(L^p\) spaces

**Measure theory**

- Borel σ-algebra, general σ-algebras
- construction of Lebesgue measure on Euclidean spaces
- outer measures, counting measure, product measures
- \(L^p\) spaces of a general measure space
- absolute continuity of measures
- Radon–Nikodym theorem

**References.**Expand to see my three favorite books on these topics.

**Other excellent sources.**

Please email comments, suggestions, and corrections to williamdemeo@gmail.com