{"product_id":"young-measures-on-topological-spaces-with-applications-in-control-theory-and-probability-theory-von-charles-castaing-paul-raynaud-de-fitte-michel-valadier","title":"Young Measures on Topological Spaces","description":"\u003cp\u003eClassicalexamples 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? ].\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9789048165520\"\u003e\u003ch3\u003eWith Applications in Control Theory and Probability Theory\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9789048165520","offer_id":39416552685661,"sku":"9789048165520","price":53.49,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/b85bcfb7-3165-483f-9e69-e89cdf85a3f8.jpg?v=1782361182","url":"https:\/\/shop.autorenwelt.de\/products\/young-measures-on-topological-spaces-with-applications-in-control-theory-and-probability-theory-von-charles-castaing-paul-raynaud-de-fitte-michel-valadier","provider":"Autorenwelt Shop","version":"1.0","type":"link"}