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Page "Følner sequence" ¶ 15
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locally and compact
In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.
Note that a locally finite Borel measure automatically satisfies μ ( C ) < ∞ for every compact set C.
The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it.
The prototypical example of a Banach algebra is, the space of ( complex-valued ) continuous functions on a locally compact ( Hausdorff ) space that vanish at infinity.
* The algebra of all bounded continuous real-or complex-valued functions on some locally compact space ( again with pointwise operations and supremum norm ) is a Banach algebra.
* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g )( h ) for x, y in L < sup > 1 </ sup >( G ).
* Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact group, where the product of two measures is given by convolution.
In spaces that are compact in this latter sense, it is often possible to patch together information that holds locally — that is, in a neighborhood of each point — into corresponding statements that hold throughout the space, and many theorems are of this character.
* Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification.
* Neither of the spaces in the previous two examples are locally compact but both are still Lindelöf
By the same construction, every locally compact Hausdorff space X is a dense subspace of a compact Hausdorff space having at most one point more than X ..
C *- algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics.
Mathematically, the conjecture states that the maximal Cauchy development of generic compact or asymptotically flat initial data is locally inextendible as a regular Lorentzian manifold.
Instead, with the topology of compact convergence, C ( a, b ) can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric.
* Compact groups or locally compact groups — Many of the results of finite group representation theory are proved by averaging over the group.
This can be done for locally compact groups, using Haar measure.
In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.
The core motivating idea are the various Fourier transforms, which can be generalized to a transform of functions defined on Hausdorff locally compact topological groups.
The theory for abelian locally compact groups is called Pontryagin duality ; it is considered to be in a satisfactory state, as far as explaining the main features of harmonic analysis goes.
For general nonabelian locally compact groups, harmonic analysis is closely related to the theory of unitary group representations.
Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support.
A normed vector space V is locally compact if and only if the unit ball B =
The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function.

locally and group
A more realistic analysis must take into account the fact that Brandywine people in the urban-fringe area are, in general, less segregated locally than group members in rural areas.
With the main aims in mind, the group launched the LocAle scheme in 2007 which was developed by Steve Westby of the Nottingham branch to help promote locally brewed beers and also help with environmental issues.
A non-compact generalization of a profinite group is a locally profinite group.
* The automorphism group of a locally finite rooted tree is profinite.
) The usual terminology is different: a group G is called locally finite if every finitely-generated subgroup is finite.
# S is locally finite and contains a copy of every finite group.
) be a locally compact topological group.
For example, the product of the unit circle ( with its usual topology ) and the real line with the discrete topology is a locally compact group with the product topology and Haar measure on this group is not inner regular for the closed subset
The terms criollo and crioulo were originally qualifiers used throughout the Spanish and Portuguese colonies to distinguish the members of an ethnic group that were born and raised locally from those who immigrated as adults.
Lebesgue measure on the real line is one of the simplest examples of a Haar measure on a locally compact group.
The East Indians, as this group was known locally, signed on for a certain number of years, after which, in theory, they would return to India with their savings from working in the sugar fields.
These are the sphere-type doughnuts and belong to a group of pastries locally known as facturas.
Johnson County officials were upset that the group was not kept locally at Ft. McKinney.
The locally organized Colorado River Concert Association is a group of local citizens who attract performing artists to the community.
Athens is in a small area locally known as ' The Valley ,' a group of four contiguous communities in the two states: Waverly, NY ; South Waverly, PA ; Sayre, PA ; and Athens.
The Tribune is locally owned and operated, while The Observer is part of the larger Houston Community Newspapers group.
As a locally compact topological group, it has a unique ( up to positive scalar multiple ) Haar measure μ.
A campaign of opposition to the impact on the town and the surrounding area of the preferred route is being locally co-ordinated by a protest group, Amersham Action Group, which along with other protest groups is part of the HS2 Action Alliance.

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