9789048165520 - young measures on topological spaces: with applications in control theory and probability theory (mathematics and its applications, band 571) von castaing, charles; raynaud de fitte, paul; valadier, michel (3 Ergebnisse)
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Taschenbuch. Zustand: Neu. Druck auf Anfrage Neuware - Printed after ordering - 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 appe…ar 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 ].
