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];
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*) =

@

@

@

@

@

@

>

**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