• Source: Vague topology

In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak-* topology which arises in the study of measures on locally compact Hausdorff spaces.
Let



X


{\displaystyle X}

be a locally compact Hausdorff space. Let



M
(
X
)


{\displaystyle M(X)}

be the space of complex Radon measures on



X
,


{\displaystyle X,}

and




C

0


(
X

)

∗




{\displaystyle C_{0}(X)^{*}}

denote the dual of




C

0


(
X
)
,


{\displaystyle C_{0}(X),}

the Banach space of complex continuous functions on



X


{\displaystyle X}

vanishing at infinity equipped with the uniform norm. By the Riesz representation theorem



M
(
X
)


{\displaystyle M(X)}

is isometric to




C

0


(
X

)

∗


.


{\displaystyle C_{0}(X)^{*}.}

The isometry maps a measure



μ


{\displaystyle \mu }

to a linear functional




I

μ


(
f
)
:=

∫

X


f

d
μ
.


{\displaystyle I_{\mu }(f):=\int _{X}f\,d\mu .}


The vague topology is the weak-* topology on




C

0


(
X

)

∗


.


{\displaystyle C_{0}(X)^{*}.}

The corresponding topology on



M
(
X
)


{\displaystyle M(X)}

induced by the isometry from




C

0


(
X

)

∗




{\displaystyle C_{0}(X)^{*}}

is also called the vague topology on



M
(
X
)
.


{\displaystyle M(X).}

Thus in particular, a sequence of measures





(

μ

n


)


n
∈

N





{\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }}

converges vaguely to a measure



μ


{\displaystyle \mu }

whenever for all test functions



f
∈

C

0


(
X
)
,


{\displaystyle f\in C_{0}(X),}






∫

X


f
d

μ

n


→

∫

X


f
d
μ
.


{\displaystyle \int _{X}fd\mu _{n}\to \int _{X}fd\mu .}


It is also not uncommon to define the vague topology by duality with continuous functions having compact support




C

c


(
X
)
,


{\displaystyle C_{c}(X),}

that is, a sequence of measures





(

μ

n


)


n
∈

N





{\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }}

converges vaguely to a measure



μ


{\displaystyle \mu }

whenever the above convergence holds for all test functions



f
∈

C

c


(
X
)
.


{\displaystyle f\in C_{c}(X).}

This construction gives rise to a different topology. In particular, the topology defined by duality with




C

c


(
X
)


{\displaystyle C_{c}(X)}

can be metrizable whereas the topology defined by duality with




C

0


(
X
)


{\displaystyle C_{0}(X)}

is not.
One application of this is to probability theory: for example, the central limit theorem is essentially a statement that if




μ

n




{\displaystyle \mu _{n}}

are the probability measures for certain sums of independent random variables, then




μ

n




{\displaystyle \mu _{n}}

converge weakly (and then vaguely) to a normal distribution, that is, the measure




μ

n




{\displaystyle \mu _{n}}

is "approximately normal" for large



n
.


{\displaystyle n.}






See also


List of topologies – List of concrete topologies and topological spaces




References



Dieudonné, Jean (1970), "§13.4. The vague topology", Treatise on analysis, vol. II, Academic Press.
G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
This article incorporates material from Weak-* topology of the space of Radon measures on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Kata Kunci Pencarian:

• Ras manusia
• Vague topology
• Weak topology
• Compact space
• List of topologies
• Operator topologies
• Dirac delta function
• Laplace transform
• Radon measure
• Distribution (mathematics)
• Spaces of test functions and distributions