Real analysis books and notes

Joshua Ruiter, Theorems: Real Analysis (March 2018) – statements and proofs covering topology, Lebesgue measure, measurable functions, integration, and later real-analysis topics.

Textbooks

Rudin, Real and complex analysis.
[StSh05]

Chapters 1–3 treat Lebesgue measure, integration, and differentiation on \(\RR^d\); Chapters 4–5 treat \(L^2\) and Hilbert spaces; Chapter 6 treats abstract measure spaces. \(L^p\) spaces for general \(p\) are treated in [Foll13] and, with convexity inequalities, in Lieb and Loss, Analysis, Chapter 2.
[StSh03]

Chapters 2 and 3 treat Fourier series, and Chapter 5 treats the Fourier transform on \(\RR\).
Evans, Partial differential equations, Chapter 5 (Sobolev spaces).
Shiryayev, Probability.
Feller, An introduction to probability theory and its applications.
Durrett, Probability: theory and examples.

McMullen, Harvard Math 212a course notes (2019)
Real analysis course notes
Dexter Chua, *Part IB — Analysis II* (Michaelmas 2015) – course notes based on lectures by N. Wickramasekera, covering uniform convergence and continuity, Riemann integration, normed and metric spaces, contraction mappings, and multivariable differentiation.

Homework and exams from UGA Math 8100, as posted by Neil Lyall:

[StSh05]

Stein, E. M. and R. Shakarchi, Real analysis: Measure theory, Integration, and hilbert spaces, Princeton University Press, 2005

[Foll13]

Folland, G. B., Real analysis: Modern techniques and Their applications, John Wiley & Sons, Incorporated, Wiley, 2013

[StSh03]

Stein, E. M. and R. Shakarchi, Fourier analysis: An introduction, Princeton University Press, 2003