{"product_id":"young-measures-on-topological-spaces-charles-castaing-9781402019630","title":"Young Measures on Topological Spaces: With Applications in Control Theory and Probability Theory","description":"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   on ? ×R, called Young measure. In Functional Analysis formulation, this is the narrow convergence to   of the image of the Lebesgue measure on ? by ? ? (?, u (?)). In the disintegrated form (  ), the parametrized measure   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(XA) 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? ].\u003cbr\u003e\u003cbr\u003e\u003cb\u003eAuthor:\u003c\/b\u003e Charles Castaing, Paul Raynaud de Fitte, Michel Valadier\u003cbr\u003e\u003cb\u003eISBN-10:\u003c\/b\u003e 1402019637\u003cbr\u003e\u003cb\u003eISBN-13:\u003c\/b\u003e 9781402019630\u003cbr\u003e\u003cb\u003ePublisher:\u003c\/b\u003e Springer\u003cbr\u003e\u003cb\u003eLanguage:\u003c\/b\u003e English\u003cbr\u003e\u003cb\u003ePublished:\u003c\/b\u003e 07\/14\/2004\u003cbr\u003e\u003cb\u003ePages:\u003c\/b\u003e 320\u003cbr\u003e\u003cb\u003eFormat:\u003c\/b\u003e Hardcover\u003cbr\u003e\u003cb\u003eWeight:\u003c\/b\u003e 1.60lbs\u003cbr\u003e\u003cb\u003eSize:\u003c\/b\u003e 9.40h x 6.40w x 0.80d","brand":"Charles Castaing","offers":[{"title":"Hardcover","offer_id":47202875965695,"sku":"9781402019630","price":54.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0662\/2982\/9887\/files\/img_3d270ab1-9a08-4d6a-8ef0-4912e889afe1.jpg?v=1756833669","url":"https:\/\/www.whiterainbookhouse.com\/products\/young-measures-on-topological-spaces-charles-castaing-9781402019630","provider":"WR Book House","version":"1.0","type":"link"}