Real Analysis and Measure Theory

Math 564, 2025 Fall

Maximum of (25% homework + 25% midterm + 50% final) and (25% homework + 75% final).
There will be roughly 6 homework assignments (one every two weeks), to be submitted on Crowdmark.
The midterm will be held in the week of Oct 21. The date of the final will be determined later by the exam office.

Polish spaces
σ-algebras and Borel sets, measurable spaces, measures and premeasures
Constructions of Bernoulli(p) measures on 2^ℕ and Lebesgue measure on ℝ^d
Carathéodory's extension theorem: outer measures and two different proofs (by C. Carathéodory and T. Tao)
Measurable and non-measurable sets
Pocket tools: increasing unions/decreasing intersections, Borel–Cantelli lemmas, measure exhaustion and application: Sierpiński's theorem for atomless measures
Approximating measurable sets: 99% lemma, measure regularity (for metric spaces) and tightness (for Polish spaces)
Applications: ergodic equivalence relations/group actions and non-measurability of transversals
Locally finite Borel measures on ℝ and increasing right-continuous functions

Measurable functions, limits of measurable functions, Luzin's theorem, push-forward measures
Push-forward measures (e.g. pushing forward a random graph to a random forest), Haar measures, measure-preserving actions and application: Poincaré recurrence theorem
Borel and measure isomorphism theorems (sketches of proofs)
Simple functions and their integration, approximation of measurable functions by simple ones
Integration of non-negative functions, monotone convergence theorem, and Fatou's lemma
Integration of real-valued functions and L¹, dominated convergence theorem, and the density of simple functions in L¹
Properties of integrable functions: σ-finiteness of support, 99% boundedness, absolute continuity of B ↦ ∫_B f dμ
(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)
Convergence in measure and relations between different modes of convergence
Product measures: finite and countable products
Fubini–Tonelli theorem and application: Kolmogorov's 0–1 law (ergodicity of eventual equality)

Measure differentiation, signed measures, Lebesgue density, and a.e. differentiable functions

Orthogonality and absolute continuity of measures, Jordan decomposition
Signed measures and Hahn decomposition (proof via measure exhaustion)
Lebesgue–Radon–Nikodym theorem and Radon–Nikodym derivatives
Lebesgue differentiation theorem for ℝ^d: proof via Hardy–Littlewood maximal function and Vitali covering lemma
Lebesgue density theorem for ℝ^d, Lebesgue differentiation theorem for all locally finite Borel measures on ℝ^d
Characterization of distribution functions on ℝ whose associated measures are absolutely continuous/orthogonal with respect to Lebesgue measure
Absolutely continuous functions and fundamental theorem of calculus for increasing functions
Finite Borel signed measures on ℝ^d and right-continuous functions of bounded variation, fundamental theorem of calculus for functions of bounded variation