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Young Measures on Topological Spaces
With Applications in Control Theory and Probability Theory, Mathematics and Its Applications 571
ISBN/EAN: 9781402019630
Umbreit-Nr.: 1662854
Sprache:
Englisch
Umfang: xii, 320 S.
Format in cm: 2 x 24.7 x 16.5
Einband:
gebundenes Buch
Erschienen am 14.07.2004
Auflage: 1/2004
- Zusatztext
- Classicalexamples of moreand more oscillatingreal-valued functions on a domain N ?of R are the functions u (x)=sin(nx)with x=(x ,.,x ) or the so-called n 1 1 n n+1 Rademacherfunctionson]0,1[,u (x)=r (x) = sgn(sin(2 ?x))(seelater3.1.4). n n They may appear as the gradients?v of minimizing sequences (v ) in some n n n?N variationalproblems. Intheseexamples,thefunctionu convergesinsomesenseto n ameasure µ on ? ×R, called Young measure. In Functional Analysis formulation, this is the narrow convergence to µ of the image of the Lebesgue measure on ? by ? ? (?,u (?)). In the disintegrated form (µ ) ,the parametrized measure µ n ? ??? ? captures the possible scattering of the u around ?. n Curiously if (X ) is a sequence of random variables deriving from indep- n n?N dent ones, the n-th one may appear more and more far from the k ?rst ones as 2 if it was oscillating (think of orthonormal vectors in L which converge weakly to 0). More precisely when the laws L(X ) narrowly converge to some probability n measure, it often happens that for any k and any A in the algebra generated by X ,.,X, the conditional law L(X A) still converges to (see Chapter 9) 1 k n which means 1 ??? C (R) ?(X (?))dP(?)?? ?d b n P(A) A R or equivalently, ? denoting the image of P by ? ? (?,X (?)), n X n (1l ??)d? ?? (1l ??)d[P? ].
- Kurztext
- Provides a unified presentation of the theory, together with new results and applications in various fields