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} *A*_{i})
= ∑_{i} *μ*(*A*_{i})
for *i* ∈ \(\mathbb N\);

* __Sigma-finite measure__:
One such that there exists a covering {*A*_{i}}_{i ∈
\(\mathbb N\)} of *X* = ∪_{i}* A*_{i}
by finite measure subsets, *μ*(*A*_{i}) < ∞ 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 R ^{n} 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].

@ __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*) =

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**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.

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send feedback and suggestions to bombelli at olemiss.edu – modified 19
may 2017