Young Measures on Topological Spaces

With Applications in Control Theory and Probability Theory

von Paul Raynaud de Fitte, Charles Castaing, Michel Valadier

320 Seiten, Hardcover

Kurzbeschreibung des Verlags

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? ].

• Themen: Mathematik und Naturwissenschaften Mathematik Optimierung
• ISBN: 9781402019630
• Sprache: Englisch
• Erscheinungsdatum: 14.07.2004
• Größe: 244 x 164 mm
• Verlag: Springer