Measure Theory| 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 : X → Y 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(A ⊔ B)
= μ(A) + μ(B) for disjoint events
A and B; A quantum measure satisfies μ(A ⊔
B ⊔ C) = μ(A ⊔ B)
+ μ(A ⊔ C) + μ(B ⊔
C) − 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 16 dec 2018