Alethograph

Infinite Dimensional Analysis

3 min read

Textbook

Authors: Charalambos D. Aliprantis, Kim C. Border | Publisher: Springer, 2006 | Edition: 3rd

Overview

A comprehensive reference on infinite-dimensional analysis covering topology, measure theory, functional analysis, and operator theory. Treats probability measures, Riesz representation theorems, and the weak* topology on spaces of measures from a functional-analytic perspective, emphasising the role of Banach lattices and AL-spaces.

Topics Studied

measure convergence (studied 22-03-2026)

Chapters read: Ch. 2 (pp. 47-55), Ch. 10 (pp. 371-401), Ch. 12 (pp. 433-452), Ch. 15 (pp. 505-523)

Key Definitions

  • Definition 10.1 (p. 374): Set functions — monotone, additive, sigma-additive, subadditive, sigma-subadditive, countably additive, finitely additive
  • Definition 10.2 (p. 374): Signed charge, charge, signed measure, measure — hierarchy of set functions
  • Definition 10.34 (p. 387): Measure space (X, Sigma, mu); probability space if mu(X) = 1; complete if Sigma = Sigma_mu
  • Definition 12.1 (p. 434): Borel measure, Baire measure, Borel charge
  • Definition 12.2 (p. 435): Tight, outer regular, inner regular, normal, regular — regularity hierarchy for charges/measures
  • Definition 12.17 (p. 443): Point mass delta_x
  • Definition 15.17 (p. 516): Borel space — metrizable space homeomorphic to Borel subset of a Polish space
  • Definition 15.20 (p. 518): Tight family — for each eps > 0, exists compact K with mu(K) > mu(X) - eps for all mu in F
  • Definition 10.59 (p. 401): Absolute continuity nu << mu

Key Theorems

  • Theorem 10.8 (p. 376): Continuity of measures — A_n increasing implies mu(A_n) increasing to mu(A); A_n decreasing with finite mu(A_k) implies mu(A_n) decreasing to mu(A)
  • Theorem 10.14, Vitali-Hahn-Saks (p. 379): Setwise convergent sequence of finite measures on common sigma-algebra defines a finite measure in the limit
  • Theorem 10.15, Dieudonne (p. 379): On Borel sets of Polish space, convergence on all open sets extends to all Borel sets
  • Theorem 10.53 (p. 397): ba(A) is an AL-space with total variation norm
  • Theorem 10.56 (p. 399): ca(A) is a projection band in ba(A); Yosida-Hewitt decomposition
  • Theorem 10.61 (p. 401): Lebesgue decomposition — unique nu = nu_1 + nu_2 with nu_1 << mu and nu_2 perp mu
  • Theorem 12.4 (p. 436): Tight charges are measures (every tight finite Borel charge on Hausdorff space is a regular measure)
  • Theorem 12.5 (p. 436): Every finite Borel measure on metrizable space is normal
  • Theorem 12.7 (p. 438): Every finite Borel measure on Polish space is regular
  • Theorem 15.1 (p. 506): U_d separates points of P(X)
  • Theorem 15.2 (p. 507): sigma(P, C_b) = sigma(P, U_d) = sigma(P, D) on P(X)
  • Theorem 15.3 (p. 508): Portmanteau — 7 equivalent characterisations of w*-convergence of nets in P(X)
  • Theorem 15.5 (p. 511): Semicontinuous functions define semicontinuous functionals on P(X)
  • Theorem 15.8 (p. 512): X embeds into P(X) via point masses; X closed in P(X) if separable
  • Theorem 15.9 (p. 512): Extreme points of P(X) are point masses (separable metrizable X)
  • Theorem 15.10 (p. 513): Density — finite-support measures dense in P(X) (metrizable X)
  • Theorem 15.11 (p. 513): X compact metrizable iff P(X) compact metrizable
  • Theorem 15.12 (p. 513): X separable metrizable iff P(X) separable metrizable
  • Theorem 15.14 (p. 514): Pushforward f-hat: P(X) → P(Y) is continuous; injective/homeomorphism for Polish injective/embedding f
  • Theorem 15.15 (p. 515): X Polish iff P(X) Polish
  • Theorem 15.18 (p. 517): X Borel space iff P(X) Borel space
  • Theorem 15.22, Prokhorov (p. 519): On Polish spaces, nonempty subset of P(X) is relatively compact iff tight

Key Examples

  • Example 12.9 (p. 439): Non-regular Borel probability measure on compact Hausdorff space Omega of ordinals — the big-small measure has no support

Atomic Notes

book

Graph View

Backlinks