Preface

These notes accompany the Fall 2024 iteration of Math 114, a second-semester undergraduate course on real analysis and measure theory. These notes deviate quite a bit from lecture materials, in particular:

  1. I prefer a more principled construction of the Lebesgue measure as the restriction of the exterior measure using the Carathéodory theorem.
    • Sets declared measurable by the Carathéodory criteria are automatically closed under \(\sigma\)-algebra operations, so one only needs to show the measurability of rectangles to establish the measurability of the Borel sets.
  2. Materials are heavily supplemented by Sheldon Axler’s Measure, Integration and Real Analysis.
  3. Some time is spent on infinite-dimensional vector spaces and results in Banach spaces, since
    1. nontrivial complexities arive when we generalize linear algebra constructions to infinite-dimensions (e.g. Hammel basis, span, and orthonormal basis).
    2. several theorems are important for convex analysis.
  4. Much more time is spent on the Fourier transform, with intuitions developed on the compact domain \(\partial D\cong S^1\) and later generalized to locally compact abelian groups.
  5. Other miscellany:
    • Proof of Young’s inequality (proposition 7.2) using the Legendre transform to highlight the role of convexity.

Organization

  1. Abstract machinery of measures and \(\sigma\)-algebra.
    • Lieb and Loss, chapter 1.
  2. The exterior measure and Lebesgue measure on \(\mathbb R^n\).
    • Hunter.
  3. Lebesgue integral, Littlewood’s three principles, and integral convergence theorems.
    • Rudin, , chapter 11; Axler, chapter 2; Lieb and Loss, chapter 1.
  4. Iterated integrals: uniqueness of product measure and Fubini’s theorem.
    • Lieb and Loss, chapter 1.
  5. Spaces of functions: normed vector spaces, dual space of bounded linear functionals,
    and the Hahn-Banach theorem; Baire’s theorem and other Banach theorems.
    • Axler, Chapter 6.
  6. \(L^p\) spaces.
    • Lieb and Loss, chapter 2; Axler, chapter 7.
  7. Fourier analysis: Example of the Fourier series on a compact domain, the Haar measure, locally compact groups and their duals.
    • Axler, Chapter 11, Rudin (Fourier analysis), Chapter 1, Su.

References for these notes

  1. Axler, Measure, Integration & Real analysis (Axler 2020).
  2. Lieb and Loss, Analysis (Lieb and Loss 2001).
  3. John K. Hunter, online notes on the Lebesgue measure.
  4. Rudin, Principles of Mathematical Analysis (Rudin et al. 1964).
  5. Rudin, Fourier analysis on groups (Rudin 2017).
  6. Dan Su, The Fourier Transform for Locally Compact Abelian Groups, online notes.

Main deliverables

  1. Monotone class theorem 2.1: freely-generated monotone class equals freely-generated \(\sigma\)-algebra for algebra of sets containing \(\Omega\).
  2. Carathéodory’s theorem 2.3: given an outer measure \(\mu\), the measurable sets \(B\subset \Omega\) satisfying \(\mu(\forall E) = \mu(E\cap B) + \mu(E\cap B^c)\) form a \(\sigma\)-algebra and obey countable additivity.
  3. Dominated convergence theorem 4.8: dominated pointwise limit of measurable functions can be exchanged with integral.
  4. Hahn-Banach, hyperplane separation theorems: for every extension of a bounded linear functional, there exists a norm-minimizing one which preserves the original norm.
  5. Fubini’s theorem
  6. Baire’s theorem 6.4: for every cover of a complete metric space by closed sets, there exists an element with nonempty interior.

Proof techniques:

  1. Working with countable collection of approximations: given tolerance \(\epsilon\), introduce approximation of index \(j\) with tolerance \(\epsilon 2^{-j}\) to have additively bounded total tolerance.
    • Examples: outer measure properties of exterior measure (proposition 3.1).
  2. Prove properties for characteristic functions, then generalize them using approximation and limit theorems here.
    • Examples: Luzin’s theorem, monotone convergence theorem 4.7,
  3. Sandwich bounds
    • Lemma 2.2, Monotone convergence theorem.
  4. Argue the convergence of a subsequence satisfying \(\sum |f_k - f_{k-1}|<\infty\) when working with Cauchy sequences.
    • Baire’s theorem 6.4. Completeness proof for \(L^p\) (proposition 7.7).
  5. Prime example for the use of monotone convergence theorem, Fatou’s lemma, and dominated convergence theorem: proposition 7.7.
  6. Using the monotone class theorem (example theorem 2.2): rewrite nested chain as disjoint union, then pass through the limit using the countable additivity of measures. Next apply the reduction from finite to \(\sigma\)-finite measures.
  7. Demonstrating a certain property (e.g. triviality): approximate by a tractable dense subset.
    • Orthonormality of the Fourier basis on \(L^2(\partial D)\) (proposition 8.7).

Mandatory todo list:

  1. Inductive generalization of Holder’s inequality.
  2. Put in some pset problems.
  3. Pset problem: separately continuous implies measurable.
  4. Dual of \(L^p\) space.

Optional:

  1. Lieb and Loss 2.9 to 2.18: compactness under weak convergence.
  2. Product measures and Fubini.
  3. Arnold-Kolmorogorov theorem.

Bibliography

Axler, Sheldon. 2020. Measure, Integration & Real Analysis. Springer Nature.
Lieb, Elliott H, and Michael Loss. 2001. Analysis. Vol. 14. American Mathematical Soc.
Rudin, Walter et al. 1964. Principles of Mathematical Analysis. Vol. 3. McGraw-hill New York.
Rudin, Walter. 2017. Fourier Analysis on Groups. Courier Dover Publications.