# 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