Measure Theory¶
Job: rigorous integration, product measures, and absolute continuity — the foundation of \(L^p\), probability, and modern analysis.
Core¶
Measure theory upgrades the Riemann integral: measurable sets/functions, monotone/dominated convergence, Fubini–Tonelli, and Radon–Nikodym supply the theorems actually used in research.
| Topic | Role |
|---|---|
| \(\sigma\)-algebras & measures | domain of integration; probability spaces \((\Omega,\mathcal{F},\mathbb{P})\) |
| Carathéodory extension | construct Lebesgue measure from outer measure |
| Integration | \(\int f\,d\mu\) for nonnegative / integrable \(f\) |
| Limit theorems | MCT, Fatou, DCT — justify lim/∫ interchange |
| Product measures | Fubini–Tonelli for \(\iint\) |
| Absolute continuity | Radon–Nikodym \(d\nu/d\mu\); Lebesgue decomposition |
| \(L^p\) | Banach spaces of integrable \(p\)-th powers |
Reading map¶
| Page | Content |
|---|---|
| Foundations | full quals package: Carathéodory through RN and \(L^p\) |
| Geometric Measure Theory | Hausdorff / dimension / fractal; rectifiability / Federer–Fleming / area formula; currents / Plateau / compactness; minimal surface regularity (De Giorgi / Almgren); varifolds / Allard / monotonicity |
| Ergodic & Recurrence | Birkhoff / maximal (Hopf–Dunford–Schwartz); Poincaré recurrence / conservative-dissipative; Kingman subadditive / Furstenberg–Kesten; Doob martingale; Besicovitch / density / differentiation |
| Probability & Stochastic | Kolmogorov extension / Wiener / Brownian; Markov / martingale; Itô integral / formula / SDE; infinite-dim / rough paths / SPDE; law / characteristic / large deviations |
| Harmonic & Singular Integrals | Hardy–Littlewood maximal / weak (1,1); Calderón–Zygmund / Cotlar / T(1) / T(b); Littlewood–Paley / Besov / Triebel; BMO / John–Nirenberg / Carleson; Muckenhoupt \(A_p\) / weighted |
Interfaces¶
- Functional analysis: \(L^p\) duals, weak convergence
- Probability: measures with \(\mu(\Omega)=1\)
- Real analysis: Riemann as special case
- Fourier analysis: \(L^1\), \(L^2\) Fourier transforms
See it¶
Construction of Brownian motion / Wiener measure needs Kolmogorov extension (consistent finite-dimensional distributions) plus Kolmogorov continuity or abstract Wiener space — all measure-theoretic. Same toolkit: \(L^2\) martingales, conditional expectation as RN derivative / projection.
Check¶
Why is completeness of the measure important for almost-everywhere statements?
Without completing Lebesgue measure, subsets of null sets need not be measurable; “\(f=g\) a.e. \(\Rightarrow\) \(f\) measurable if \(g\) is” can fail. Complete \(\sigma\)-algebras close this gap.
Does \(f_n\to f\) a.e. and \(\sup\int|f_n|<\infty\) imply \(L^1\) convergence?
No — classic moving bump \(f_n=n\cdot 1_{[0,1/n]}\): \(\int f_n=1\), \(f_n\to 0\) a.e., not UI; DCT needs domination by an integrable \(g\).
Pitfalls¶
- Almost everywhere equal functions define the same \(L^p\) class — representatives still matter for pointwise evaluation.
- Infinite measure spaces: \(L^p\not\subseteq L^q\) in general (compare finite vs infinite \(\mu(X)\)).
- Product \(\sigma\)-algebra vs completion of product measure — Fubini needs care with null sets in each variable.
Refs¶
- Folland, Real Analysis
- Rudin, Real and Complex Analysis
- Bogachev, Measure Theory
Next¶
Foundations · Geometric Measure Theory · Ergodic & Recurrence · Probability & Stochastic · Harmonic & Singular Integrals · Functional analysis · Probability · Real analysis