Measure Theory

In General > s.a. Borel Measure; distance and types of distances; vectors [inner product].
$Measurable space: A set X with a sigma field of subsets (> see ring).$ Measure space: A measurable space and a function μ, which is (1) Positive, μ: $$\cal A$$ → $$\mathbb R$$1 ∪ {+∞}, where $$\cal A$$ is the σ-field of subsets; (2) Countably additive, μ(∪i Ai) = ∑i μ(Ai) for i ∈ $$\mathbb N$$;
* Sigma-finite measure: One such that there exists a covering {Ai}i ∈ $$\mathbb N$$ of X = ∪i Ai by finite measure subsets, μ(Ai) < ∞ for all i.
* Measurable map: A map f : XY is measurable iff the pre-image of every measurable set in Y is a measurable set in X; Notice that if the measures in question are Borel measures, all continuous maps are measurable.

Measure on a Group
* Haar measure: A left-invariant regular Borel measure on a (locally connected) Lie group; It is guaranteed to exist (on a locally connected Lie group); It is unique on a compact group, and determined up to a global factor on a non-compact group.
@ Haar measure: Creutz JMP(78) [on SU(N)]; Sun et al ht/01 [non-existence on C($$\mathbb R$$n, U(1))]; Niemiec T&A(08) [generalization]; > s.a. lie groups.
@ On special groups: Sun et al ht/01 [on C($$\mathbb R$$n, G)]; > s.a. examples of lie groups; Virasoro Group.

On Rn or Integers > see Central Limit Theorem; Gamma, gaussian, Poisson Distribution.

On Other Sets > s.a. boundaries [measure on boundary conditions]; connection; lie group; loop; posets [partially ordered measure spaces].
* On a set of paths: The Wiener measure, introduced in the theory of brownian motion, amounts to approximating each path as being piecewise linear, performing a finite number of integrations over the intermediate positions and taking a limit.
* On a set of evolutions: If we identify the set of dynamical evolutions with the (covariant) phase space, we can use the Liouville measure.
@ Wiener measure: Stroock & Varadhan 79; Choquet-Bruhat et al 82, p583; Andersson & Driver JFA(99) [approximations]; Jiang JSP(14) [relationship between two types of Wiener measures]; Belokurov & Shavgulidze a1812 [polar decomposition, calculations].
@ Other measures on paths: Durhuus & Jonsson mp/00; Betz et al mp/04 [Gibbs measures on Brownian paths]; > s.a. loops.
@ Sets of transformations: Niemiec T&A(06) [equicontinuous semigroups of continuous transformations of a compact Hausdorff space].
@ Infinite-dimensional linear spaces: Gel'fand & Vilenkin 64; Yamasaki 85; > s.a. path integrals.
@ Other infinite-dimensional sets: Ashtekar & Lewandowski JMP(95) [connections]; Menotti & Peirano NPB(97)ht/96 [Euclidean metrics]; Djah et al mp/04 [functional, Feynman-graph representation]; Vershik mp/07 [Lebesgue measure]; > s.a. Prevalence; random fields.

And Physics > s.a. probability in physics.
* Classical vs quantum measures: A classical measure satisfies μ(AB) = μ(A) + μ(B) for disjoint events A and B; A quantum measure satisfies μ(ABC) = μ(AB) + μ(AC) + μ(BC) − μ(A) − μ(B) − μ(C) for disjoint events A, B and C.
@ For classical systems: Werner JMP(11) [dynamically defined measures and equilibrium states].
@ On the space of mixed quantum states: Życzkowski & Sommers JPA(01)qp/00.
@ (Positive) operator-valued measures: in Holevo 82; in Busch et al 95; Cassinelli et al qp/02, JMP(03)qp [and group action]; Skulimowski PLA(02), PLA(02) [and time]; Gazeau & Heller a1408 [POVM quantization].
@ On projection lattices of von Neumann algebras: Hamhalter 03.
@ Quantum measures: Sorkin MPLA(94)gq, in(97)gq/95; Salgado MPLA(02)gq/99; Zafiris JMP(06) [sheaf-theoretic representation]; Craig et al JPA(07)qp/06 [and Bell inequality analog]; Sorkin JPA(07)qp/06; Barnett et al JPA(07) [Popescu-Rohrlich boxes]; Dowker & Ghazi-Tabatabai JPA(08)-a0712 [dynamical wave-function-collapse models]; Surya & Wallden FP(10)-a0809 [quantum covers]; Ghazi-Tabatabai PhD(09)-a0906 [new interpretation]; Gudder JMP(09)-a0909 [and integration theory], RPMP(11)-a1005 [and coevent interpretation]; Dowker et al JPA(10)-a1007; Gudder a1009-proc [and quantum computers]; Gudder a1011 [Hilbert-space representation]; Sorkin a1104-in; Gudder RPMP(12) [and integrals]; Xie et al FP(13) [super quantum measures on finite spaces]; Joshi et al IJQI(16)-a1308 [higher-order, no-signaling violation]; Boës & Navascués PRD(17)-a1609 [composing decoherence functionals]; Mozota & Sorkin IJTP(17)-a1610 [determining μ(E) experimentally]; > s.a. integration [quantum integrals]; random walk [quantum].
@ Related topics: Markowich et al JFA-a0911 [Bohmian interpretation and phase-space measures].
> Specific areas of physics: see multiverse; foundations of quantum mechanics; path integrals [including coevent formulation]; quantum gauge theory.

Other References > s.a. functional analysis.
@ General: Kolmogorov & Fomin 61; Halmos 74; Swartz 94 [and function spaces]; Cohn 13 [III]; Weaver 13 [III, and functional analysis].
@ And probability: Doob 94; Adams & Guillemin 96.
@ Books, special emphasis: Parthasarathy 67 [probability measures]; Chaumont & Yor 12 [problems, r CP(13)].
@ Geometric: Federer 69; Klain & Rota 97.
@ Spaces of measures: Busch LMP(98)mp [orthogonality and disjointness]; > s.a. types of metrics.
@ Related topics: Eckmann et al Nonlin(00)cd/99, cd/99-conf [porosity and dimension]; Finster JraM(10)-a0811 [causal variational principles].
> Online resources: David Fremlin's book.