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 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 .
- 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|
|Date of creation||2013-03-22 15:22:58|
|Last modified on||2013-03-22 15:22:58|
|Last modified by||stevecheng (10074)|