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.