MATH-6200 Real Analysis

Reading List

All the references are to the textbook R. M. Dudley, Real Analysis and Probability, Cambridge University Press 2002. “Up to” will always mean “up to and including”. Unless indicated otherwise, you can ignore all the passages in the book marked by an asterisk (*). All the reading suggestions give only a sufficient minimum; you are welcome to explore other nooks and crannies on your own.

• Elementary set theory. Sections 1.1, 1.2, 1.4.
• General topology.
1. Section 2.1 (up to Definition of continuity on p. 28),
2. Sec. 2.2 (pp. 34-35), Sec. 2.3,
3. Sec. 2.5 (p. 58, actually I prefer the proof outlined in Notes to Section 2.5, please read it).
• Measures: sigma-algebras, rings and semirings, completion, Lebesgue measure.
1. Sec. 3.1. You can skip all proofs except for the proof of Theorem 3.1.4, which introduces and employs an important notion of the outer measure. Also the simple concept discussed below Lemma 3.1.2 will be used later on.
2. Sec. 3.2. Up to Example on p. 99; skip or skim the proofs.
3. Sec. 3.3 (pp. 101-102).
4. Sec. 3.4 (note that in class we discussed a nonmeasurable set on the unit circle, similar to the one constructed in Theorem 3.4.4).
• Integration. Sec. 4.1.
• Convergence theorems for integrals. Sec. 4.3.
• Product measures. Fubini's theorem. Sec. 4.4.
• Basic probability. Infinite products of probability spaces. Sections 8.1-8.2.
• L^P Spaces. Sections 5.1-5.2.
• Hilbert spaces, orthonormal bases. Sec. 5.3.
• Relations between two measures: Radon–Nikodym theorem. Sec. 5.4.
• Laws of large numbers.
• Distribution functions and densities.
• Characteristic functions (Fourier transform).
• Central limit theorem.

RPI Math Home Page