C. Castaing, M. Valadier,Convex analysis and measurable multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....only consider the case m = 1. That is, we assume that (4) is ful lled for m = 1. Since we intend to prove convergence in probability we can suppose that the random variables x 1 ( x 2 ( are contained in B( Such random variables exist because B is a random set, see Caistaing and Valadier [5] Chapter III. Indeed, B is forward absorbing such that (t; x i ( 2 B( t ) with probability 1 for any 0 if t is suciently large. Let w(t; be de ned by (t; x 1 ( t; x 2 ( Since k k V = kA kH , we obtain by (3) dkwk 2ckwk V 2l( t; x 1 ( ....

C. Castaing and M. Valadier. Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580. Springer, New York, 1977.

....inequality also holds taking t 0 Gamma and since k is arbitrary we get: r (x) h x (x; y) 4) By 2) and Rademacher s Theorem is differentiable a.e. in Omega Gamma On the other, hand since Y is compact and separable and using 1) h admits a measurable selection say s (see [6] or the measurable selection Theorem of Brown and Purves in [25] If is differentiable at x 2 Omega and y 2 h (x) then by 3) we get: h x (x; y) h x (x; s (x) so that with (3) h (x) fs (x)g: Corollary 1 Let 1 and 2 be h convex and finite, if for Gammaa.e. x ....

C. Castaing, M. Valadier. Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, vol. 580, Springer-Verlag (1977).

....k 1 R Omega g 1 1I jfjk f dR by dominated convergence. The uniqueness follows. More about duality in Orlicz spaces. In the case where is non finite, i.e. t o ) 1 for some t o 0; M = f0g and the above proof breaks down. A representation result for L 0 1 is wellknown (see for instance [CaV], Theorem VIII.5) This corresponds to a non finite function with (t) 0; 80 t t 1 ; for some t 1 0: It doesn t seem that a representation result already appeared in the literature in the alternate case where is non finite, t) 0 , t = 0 and R( Omega Gamma = 1: Our aim in this ....

....state in Theorem 4. 7 a representation result for L 0 with a general Young function : More precisely, we are going to describe the space L b : This result is an extension of the result of representation of L 0 1 : Our proof is partly an adaptation of classical proofs on the subject (see [CaV], Chapter VIII) Lemma 4.4. The space L is a band in L b : This means that (a) for any 2 L b ; f 2 L ; f 0; j j f ) 2 L and (b) for any nonempty H ae L such that there exists 2 L b which is greater than any element of H; then the supremum of H in L b ....

C. Castaing and M. Valadier. Convex analysis and measurable multifunctions. Lecture Notes in Mathematics 580. (1977), Springer, Berlin.

....the restriction of U to Omega ffi Theta R m is a continuous function. Proof of Theorem 4. 4 Note that there exists a sequence Omega k of compact subsets of Omega such that meas( Omega n Omega k ) 0, k 1, and the restriction of K to Omega k is continuous in the Hausdorff metric, cf. [CV]. Fix k 2 N and fix a Lebesgue point x 0 of Omega k . We assert that there exists a sequence u k 2 l A W 1;1 0 ( Omega Gamma R m ) such that dist(Du k ( Delta) K(x 0 ) 0 in L 1 as k 1: In fact by (4.5) for each ffl 0 we can find a function OE ffl 2 l A W 1;1 0 (B(x 0 ; ffl) ....

Castaing C., Valadier M.. Convex analysis and measurable multifunctions. Lecture Notes in Mathematics, vol. 580. Springer-Verlag, BerlinNew York, 1977.

....we only consider the case m = 1. That is, we assume that (4) is ful lled for m = 1. Since we intend to prove convergence in probability we can suppose that the random variables x 1 ( x 2 ( are contained in B( Such random variables exist because B is a random set, see Caistaing and Valadier [5] Chapter III. Indeed, B is forward absorbing such that (t; x i ( 2 B( t ) with probability 1 for any 0 if t is suciently large. Let w(t; be de ned by (t; x 1 ( t; x 2 ( Since k k V = kA 1=2 kH , we obtain by (3) dkwk 2 H dt 2ckwk 2 V 2l( t; x ....

C. Castaing and M. Valadier. Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580. Springer, New York, 1977.

....W z = d Txg if Ax = b and x 0, and the empty set otherwise. The set valued map Gamma 1 may have empty values, but it certainly has a closed graph, hence Gamma 1 is measurable but moreover, the inverse Gamma Gamma 1 (C) of a compact set C is a closed set (see e.g. Castaing and Valadier [6] for the definitions) Define now Gamma(x; Gamma 1 Gamma x; W ( d( T ( Delta : By the measurability of the data in (5.2) the set valued map Gamma is measurable jointly in (x; this since the inverse Gamma Gamma (C) of a compact set is the inverse of the closed set Gamma ....

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580, Springer-Verlag, Berlin, 1977.

....on the corresponding measurable spaces. It is also obvious that the results reported in this paper can be restated for a finite measure space. The correspondence F is said to be measurable if its graph (t, x) # T X : x # F (t) belongs to the product # algebraT# B(X) See Chapter III of [11] for many equivalent definitions of the measurability of a correspondence. A function f from (T,T ,#)toXis called a selection of F if f(t) # F (t) for all t # T . If, in addition, f is measurable, then f is said to be a measurable selection. Note that if F is measurable, then F has a ....

....known for the case of normcompact valued measurable correspondences in the separable case. Next note that even if X is separable, there is little information about Michael s topology on F(X) For example, it is not even known how the usual notion of closed valued measurable correspondences (see [4, 11], and [20] is related to the notion of RRCSs. Also the larger saturation imposed on the nonstandard model was used there to work with topologies on F(X)andX, which excludes the natural hyperfinite Loeb counting spaces obtained from the ultrapower construction on a free ultrafilter on N. Finally ....

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C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580, Springer-Verlag, Berlin and New York, 1977. MR 57:7169

....operations of mathematical morphology and stochastic geometry. It generates the myopic topology for compact sets in IR d and a modification generates the hit or miss topology for closed sets in IR d . For explanation and discussion see [39, pp. 63 92] for proofs [27, p. 15] 51, thm 3. 1] [5, 6, 11]. The discussion in [39] makes it clear that continuity with respect to H is a very desirable property for image processing algorithms. However H is never used in practice (to the author s knowledge) as an error measure for images. It has been used to measure differences of sets in functional ....

C. Castaing and M. Valadier. Convex analysis and measurable multifunctions. Lecture Notes in Mathematics 580. Springer, 1977.

....again A (r) A Phi D r where D r is the ball of radius r in the given metric. Interpretation (7) connects Hausdorff distance with the partial order of set inclusion. 2. 2 Myopic topology Topologies on spaces of subsets were introduced by Michael [36] Fell [20] and Matheron [35] See also [6, 7, 15, 21, 33, 71]. Related topologies have been constructed for uppersemicontinuous functions [52, 68] Radon measures [10, 31] and capacities in general [39, 40, 41, 42, 43, 69, 70] The present section collects definitions and important facts from the above and [3] Definition 5 The myopic topology on K 0 ....

C. Castaing and M. Valadier. Convex analysis and measurable multifunctions. Lecture Notes in Mathematics 580. Springer, 1977.

....be shown that in this setting, a random lsc function f is completely identified by a countable collection of extended real valued random variables f f x;ae j x 2 R; ae 2 Q j g where R is a countable dense subset of X. As in x2, let s begin with some well known properties of random sets [1, 4]. Propositions 3.2 and 3.3 follow directly from the definitions, X Polish, and a not so surprising, but nontrivial, projection result: Theorem III.23 of [1] Theorem 3.1 (measurable projection theorem) Suppose S is P complete and G is an S Omega B measurable subset of Xi Theta X. Then, prj ....

....x;ae j x 2 R; ae 2 Q j g where R is a countable dense subset of X. As in x2, let s begin with some well known properties of random sets [1, 4] Propositions 3.2 and 3. 3 follow directly from the definitions, X Polish, and a not so surprising, but nontrivial, projection result: Theorem III.23 of [1]. Theorem 3.1 (measurable projection theorem) Suppose S is P complete and G is an S Omega B measurable subset of Xi Theta X. Then, prj Xi G 2 S, i.e. the projection of G on Xi is S measurable. Proposition 3.2. Suppose S : Xi X is a random set and for all 2 Xi, let cl S( cl ....

C. Castaing and M. Valadier. Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580. Springer, 1977.

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C. Castaing, M. Valadier,Convex analysis and measurable multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin 1977.

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Castaing, C. and M. Valadier. Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580. Berlin: Springer-Verlag 1977

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Castaing, C. and Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580. Berlin: Springer 1977

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C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin, 1977.

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Castaing, C. and M. Valadier. Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580. Berlin: Springer-Verlag 1977

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Castaing, C. and Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580. Berlin: Springer 1977

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C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin, 1977.

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