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 
,
which is (1) Positive, :
→ R1
{+
}, where is
the 
-field
of subsets; (2) Countably additive, (i Ai)
= 
i (Ai)
for i 
N;
* Sigma-finite measure:
One such that there exists a covering {Ai}i in
N of X = i Ai
by finite measure subsets, (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.
Spaces of Measures > s.a. types
of metrics.
@ References: Busch LMP(98)mp [orthogonality and disjointness].
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 Cinfty(Rn,
U(1))]; Niemiec T&A(08) [generalization]; > s.a. lie
groups.
@ On special groups: Sun et al ht/01 [on
Cinfty(Rn, 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;
quantum
gauge theory.
* 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.
@ Paths: Strook & Varadhan 79, Choquet-Bruhat et al 82, p583
[Wiener measure]; Andersson & Driver JFA(99)
[Wiener measure, approximations]; 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.
@ Space of mixed quantum states: Zyczkowski & Sommers JPA(01)qp/00.
And Quantum Theory > s.a. foundations
of quantum mechanics; path
integrals; probability in physics.
@ (Projection) 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].
@ On projection lattices of von Neumann algebras: Hamhalter 03.
@ Quantum measures: Sorkin 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]; Barnett et al JPA(07)
[Popescu-Rohrlich boxes]; Surya & Wallden a0809 [quantum
covers]; Ghazi-Tabatabai a0906-PhD
[new
interpretation]; Gudder a0909 [and integration theory].
@ Related topics: Markowich et al a0911 [Bohmian
interpretation and phase-space
measures].
Other References
@ General: Kolmogorov & Fomin 61; Halmos 74; Doob 94 [and probability];
Adams & Guillemin
96 [and probability].
@ Books, special emphasis: Parthasarathy 67 [probability measures].
@ Geometric: Federer 69; Klain & Rota 97.
@ Related topics: Eckmann et al cd/99, cd/99-in
[porosity and dimension].
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send feedback and suggestions to bombelli at olemiss.edu – modified 11
nov 2009