McGill

Real Analysis and Measure Theory

Real Analysis and Measure Theory

Math 564, 2025 Fall

Class info

Instructor

 integration

Lectures

Homework

 Cantor function

Course material

Assessment

Topics

  1. Measures, their construction and properties
    1. Polish spaces
    2. σ-algebras and Borel sets, measurable spaces, measures and premeasures
    3. Constructions of Bernoulli(p) measures on 2ℕ​​ and Lebesgue measure on ℝ​​d
    4. Carathéodory's extension theorem: outer measures and two different proofs (by C. Carathéodory and T. Tao)
    5. Measurable and non-measurable sets
    6. Pocket tools: increasing unions/decreasing intersections, Borel–Cantelli lemmas, measure exhaustion, and Sierpiński's theorem for atomless measures
    7. Approximating measurable sets: 99% lemma, measure regularity (for metric spaces) and tightness (for Polish spaces)
    8. Applications: ergodic equivalence relations/group actions and non-measurability of transversals
  2. Measurable functions and integration
    1. Measurable functions, limits of measurable functions, approximating measurable functions with Borel and continuous functions (Luzin's theorem)
    2. Push-forward measures, Haar measures, measure-preserving actions and application: Poincaré recurrence theorem
    3. Borel and measure isomorphism theorems (sketches of proofs)
    4. Simple functions and their integration, approximation of measurable functions by simple ones
    5. Integration of non-negative functions, monotone convergence theorem, and Fatou's lemma
    6. Integration of real-valued functions and L1, dominated convergence theorem, and the density of simple functions in L1
    7. Properties of integrable functions: σ-finiteness of support, 99% boundedness, absolute continuity of B ↦​​ ∫B f dμ
    8. (Optional) Applications: Birkhoff's pointwise ergodic theorem, an elementary proof of it, and applications of this theorem (including Kolmogorov's strong law of large numbers)
    9. Convergence in measure and relations between different modes of convergence
    10. Product measures: finite and countable products
    11. Fubini–Tonelli theorem and application: Kolmogorov's 0–1 law (ergodicity of eventual equality)
  3. Measure differentiation, signed measures, Lebesgue density, and a.e. differentiable functions
    1. Orthogonality and absolute continuity of measures, Jordan decomposition
    2. Signed measures and Hahn decomposition (proof via measure exhaustion)
    3. Lebesgue–Radon–Nikodym theorem and Radon–Nikodym derivatives
    4. Lebesgue differentiation theorem for ℝ​​d: proof via Hardy–Littlewood maximal function and Vitali covering lemma
    5. Lebesgue density theorem for ℝ​​d, Lebesgue differentiation theorem for all locally finite Borel measures on ℝ​​d
    6. Locally finite Borel measures on ℝ​​ and increasing right-continuous functions
    7. Characterization of distribution functions on ℝ​​ whose associated measures are absolutely continuous/orthogonal with respect to Lebesgue measure
    8. Absolutely continuous functions and fundamental theorem of calculus for increasing functions
    9. Finite Borel signed measures on ℝ​​ and right-continuous functions of bounded variation, fundamental theorem of calculus for functions of bounded variation