Older versions of Qiuzhen Lectures on Analysis
Revision after this version: A major revision was made to Ch. 21 on Hilbert spaces. Sec. 27.5-27.7 on spectral theory was completely rewritten and moved to Ch. 25. I hope this new presentation of Riesz’s original treatment of the spectral theorem provides better insight into the monotone convergence extension, an important method in measure theory. The name of Ch. 27 was also changed to “completeness and duality in measure theory”.
Revision after this version: The order of Sections 25.5 and 25.6 has been swapped. Additionally, the section “Regularity beyond finite measures” (now Section 25.6, previously Section 25.5) is now a starred section. This change reflects my view that regularity is not particularly useful for sets with infinite outer measures. Instead, the most effective approach when dealing with such sets is to reduce the problem to sets with finite outer measures, such as the proof of Fubini’s theorem and the proof of the criterion for Radon measures in Section 25.5 (formerly Section 25.6).