weak-* topology of the space of Radon measures
Let be a locally compact Hausdorff space.
Let denote the space of complex Radon measures
on , and
denote the dual of the , the complex-valued continuous functions
on
vanishing at infinity, equipped with the uniform norm.
By the Riesz Representation Theorem
, is isometric to ,
The isometry maps a measure
into the linear functional
.
The weak-* topology (also called the vague topology) on ,
is simply the topology of pointwise convergence of :
if and only if
for each .
The corresponding topology on induced by the isometry from is also called
the weak-* or vague topology on . Thus one may talk about “weak convergence” of measures
. One of the most important applications of this notion is in probability theory:
for example, the central limit theorem is essentially the statement that
if are the distributions
for certain sums of independent random variables
.
then converge
weakly to a normal distribution
,
i.e. the distribution is “approximately normal” for large .
References
- 1 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
Title | weak-* topology of the space of Radon measures |
---|---|
Canonical name | WeakTopologyOfTheSpaceOfRadonMeasures |
Date of creation | 2013-03-22 15:22:58 |
Last modified on | 2013-03-22 15:22:58 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 4 |
Author | stevecheng (10074) |
Entry type | Example |
Classification | msc 46A03 |
Classification | msc 28A33 |
Related topic | LocallyCompactHausdorffSpace |