{"product_id":"young-measures-on-topological-spaces-with-applications-in-control-theory-and-probability-theory-9789048165520","title":"Young Measures on Topological Spaces: With Applications in Control Theory and Probability Theory","description":"\u003cp\u003e • Author(s): Charles Castaing\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Springer\u003cbr\u003e • BISAC: Mathematical Analysis\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\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   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? ].\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Paperback","offer_id":45282859909271,"sku":"9789048165520","price":3672.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9789048165520.webp?v=1769303394","url":"https:\/\/atlanticbooks.com\/products\/young-measures-on-topological-spaces-with-applications-in-control-theory-and-probability-theory-9789048165520","provider":"Atlantic Books","version":"1.0","type":"link"}