Mathematics Honors Courses in Armenia

Initiative of Armenian Society of Fellows (ASOF)

Measure theory with ergodic horizons

2025 Spring

Lecture info

Problem session info

Details

 Cantor function

Lectures

Homework

 Cantor function

Prerequisites

Absolute comfort with definitions and proofs, as well as familiarity with basic real analysis and abstract metric spaces (firm knowledge of open/closed sets, compactness, convergence, continuity, etc).

Coursework and grades

Literature

Topics

  1. Measures, their construction and properties
    1. Polish spaces, σ-algebras and Borel sets, measurable spaces
    2. Measures and premeasures
    3. Constructions of Bernoulli(p) measures on 2ℕ​​ and the 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 application: Sierpiński's theorem for atomless measures
    7. Borel measures: their regularity (for metric spaces) and tightness (for Polish spaces), the 99% lemma for Bernoulli(p) and Lebesgue measures
    8. Applications: ergodic group actions/equivalence relations and non-measurability of transversals
    9. Locally finite Borel measures on ℝ​​ and increasing right-continuous functions
  2. Measurable functions and integration
    1. Measurable functions, Luzin's theorem, push-forward measures, and random objects such as random graphs
    2. 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/complex valued functions and L1, dominated convergence theorem, and the density of simple functions in L1
    7. Properties of integrable functions: σ-finiteness of the support, 99% boundedness, absolute continuity of B ↦​​ ∫B f dμ
    8. 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 and the Fubini–Tonelli theorem
    11. Application: Kolmogorov's 0–1 law (ergodicity of eventual equality)
  3. Measure differentiation, 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. The Lebesgue-Radon-Nikodym theorem and Radon-Nikodym derivatives
    4. The Lebesgue differentiation theorem for ℝ​​d: proof via the Hardy–Littlewood maximal function and the Vitali covering lemma
    5. The Lebesgue density theorem for ℝ​​d, the Lebesgue differentiation theorem for all locally finite Borel measures on ℝ​​d
    6. Characterization of the distribution functions on ℝ​​ whose associated measures are absolutely continuous/orthogonal with respect to Lebesgue measure
    7. Absolutely continuous functions and the fundamental theorem of calculus for increasing functions
    8. Finite Borel signed measures on ℝ​​d and right-continuous functions of bounded variation, the fundamental theorem of calculus for functions of bounded variation