R. J. Aumann. Integrals of set-valued functions. J. Math. Anal. Appl., 12:1--12, 1965.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....4, the infinite horizon FEDR from a stationary policy is given as a fixed point of a corresponding operator, which is used to obtain the optimality equation and characterize a Pareto optimal policy in Section 5. 2. Preliminaries We write fuzzy sets on R by their membership functions [0, 1] (see Novak [13] and Zadeh [20] The # cut (# [0, 1] of the fuzzy set s on R defined as s # : x # # (# 0) and s 0 : cl x s(x) 0 , where cl denotes the closure of the set. A fuzzy set s is called convex if s(#x (1 #)y) s(y) x, y , # where a b ....

....is given as a fixed point of a corresponding operator, which is used to obtain the optimality equation and characterize a Pareto optimal policy in Section 5. 2. Preliminaries We write fuzzy sets on R by their membership functions [0, 1] see Novak [13] and Zadeh [20] The # cut (# [0, 1]) of the fuzzy set s on R defined as s # : x # # (# 0) and s 0 : cl x s(x) 0 , where cl denotes the closure of the set. A fuzzy set s is called convex if s(#x (1 #)y) s(y) x, y , # where a b = min a, b . Note that s is convex if and only if ....

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Aumann, R. J., Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12. 11

....for the Young measure so that the equivalent probability measure is carried by the subset L F of L R d , consisting of all integrable selectors of a given multifunction F . When this result is applied barycentrically to a suitable Young measure, it causes Aumann s well known identity [5] for the convex hull of the integral of the multifunction F to follow by a well known fact from probability theory: A vectorvalued random variable which takes its values in a convex set, has its expectation belong to that same set. A slight generalization of Aumann s result is also obtained in ....

....multifunction F . Now observe that L F [L F ] consists precisely of all measurable functions f 2 L R d which satisfy JF (f) 1 [respectively, J gF (f) 1] Thus, it follows from the definition of the oe algebra that L F and L F are measurable in L R d . Following Aumann [5], the integral of the multifunction F is defined by F d : f f d : f 2 L F g: Hence, F d is the image of L F under OE 7 OE d; note for future use that this mapping is measurable, as is evident from writing OE d = max(OE ; 0)d Gamma max( GammaOE ; 0)d, i = ....

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Aumann, R.J. (1965). Integrals of set-valued functions. J. Math. Anal. Appl. 12 1-12.

....closed set assumed to take values in the space K of all compact subsets of E . In this section we recall several notions of the expectation for random compact sets in R . The first is due to Artstein and Vitale [2] It is called the Aumann expectation, since this concept appeared implicitly in [3]. A random vector 2 R is said to be a selection of X if 2 X with probability one. Then the Aumann expectation of X is defined as EX = fE : is a selection of X , E existsg : We write kxk for the norm of x 2 R . Then kXk = sup fkxk: x 2 Xg is a random variable. The condition E kXk 1 ....

....x 2 Xg is a random variable. The condition E kXk 1 is enough to ensure that EX is nonempty and compact. If the basic probability space (used to define X) is non atomic, then EX is convex, and, moreover, EX = E conv(X) even if X is deterministic) where conv(X) is the convex hull of X , see [3, 25]. For instance, E f0; 1g = 0; 1] Alternatively, it is possible to define the Aumann expectation through the support function: h(X; u) sup fhu; xi: x 2 Xg ; of X , where u runs over the unit sphere S , and hu; xi is the inner product of u with x. Then EX can be defined as the convex set ....

R.J. Aumann. Integrals of set-valued functions. J. Math. Anal. Appl., 12:1--12, 1965.

....that is homeomorphic to a complete separable metrisable space. The Borel sigma algebra of X is denoted by B(X) We will mainly be working in a space E Theta X where X is a Polish space. The canonical projection of E Theta X onto E is denoted by pr. If A 2 E Omega B(X) then pr(A) 2 E , see [Au65] and [D72] Furthermore there is a countable family (f n ) n1 of measurable functions fn : pr(A) X such that (1) for each n 1 the graph of fn is a selection of A, i.e. f(j; fn (j) j j 2 pr(A)g ae A, 2) for each j 2 pr(A) the set ffn (j) j n 1g is dense in A j = fx j (j; x) 2 Ag. We call ....

R. Aumann, Integrals of Set-Valued Functions, Journal of Mathematical Analysis and Applications 12 (1965), 1--12.

....2 Lh 2 : Note that if F is defined in the finite interval [0; N ] then the estimate near the boundary of the interval is O(h) This follows from the corresponding result in the scalar case . Before concluding the paper, we state a conjecture that was inspired by the famous theorem of R. Aumann [1] stating that the integral of a compact valued multimap is a convex set. Since the Riemann integral is the limit of Riemann sums, which are in essence averages with positive weights, we expect that the repeated application of (8) with the spline weights generates in the limit a convex valued ....

R. J. Aumann,Integrals of set-valued functions, J. of Math. Analysis and Applications, 12 (1965) 1--12.

.... Then E(sX ( Delta) is again a support function, namely that of a deterministic convex set EX , which is called the Aumann expectation of X , see e.g. 21] Note that it is possible to define this expectation also in terms of selections or through the integral of a multivalued function, see [1], 22] and [21] The corresponding variance is Z S d Gamma1 E(E(sX (t) Gamma s X (t) 2 dt : 4 It is possible to interpret the Aumann mean as a Bochner integral in the Banach space of continuous functions on S d Gamma1 . The Fr echet approach for the space L 2 (S d Gamma1 ) yields ....

R.J. Aumann, "Integrals of set-valued functions," J. Math. Anal. Appl. 12, pp. 1-12, 1965.

....for the Young measure so that the equivalent probability measure is carried by the subset L 1 F of L 0 R d , consisting of all integrable selectors of a given multifunction F . When this result is applied barycentrically to a suitable Young measure, it causes Aumann s well known identity [5] for the convex hull of the integral of the multifunction F to follow by a well known fact from probability theory: A vectorvalued random variable which takes its values in a convex set, has its expectation belong to that same set. A slight generalization of Aumann s result is also obtained in ....

....F . Now observe that L 0 F [L 1 F ] consists precisely of all measurable functions f 2 L 0 R d which satisfy J F (f) 1 [respectively, J gF (f) 1] Thus, it follows from the definition of the oe algebra that L 0 F and L 1 F are measurable in L 0 R d . Following Aumann [5], the integral of the multifunction F is defined by Z Omega F d : f Z Omega f d : f 2 L 1 F g: Hence, R Omega F d is the image of L 1 F under OE 7 R Omega OE d ; note for future use that this mapping is measurable, as is evident from writing R OE i d = R max(OE i ....

[Article contains additional citation context not shown here]

Aumann, R.J. (1965). Integrals of set-valued functions. J. Math. Anal. Appl. 12 1-12.

....space that is homeomorphic to a complete separable metrisable space. The Borel sigma algebra of X is denoted by B(X) We will mainly be working in a space E Theta X where X is a Polish space. The canonical projection of E Theta X onto E is denoted by pr. If A 2 E Omega B(X) then pr(A) 2 E , see [Au65] and [D72] Furthermore there is a countable family (f n ) n1 of measurable functions fn : pr(A) X such that (1) for each n 1 the graph of fn is a selection of A, i.e. f(j; fn (j) j j 2 pr(A)g ae A, 2) for each j 2 pr(A) the set ffn (j) j n 1g is dense in A j = fx j (j; x) 2 Ag. We call ....

R. Aumann, Integrals of Set-Valued Functions, Journal of Mathematical Analysis and Applications 12 (1965), 1--12.

....closed set assumed to take values in the space K of all compact subsets of E . In this section we recall several notions of the expectation for random compact sets in R m . The first is due to Artstein and Vitale [2] It is called the Aumann expectation, since this concept appeared implicitly in [3]. A random vector 2 R m is said to be a selection of X if 2 X with probability one. Then the Aumann expectation of X is defined as EX = fE : is a selection of X , E existsg : We write kxk for the norm of x 2 R m . Then kXk = sup fkxk: x 2 Xg is a random variable. The condition E ....

....x 2 Xg is a random variable. The condition E kXk 1 is enough to ensure that EX is nonempty and compact. If the basic probability space (used to define X) is non atomic, then EX is convex, and, moreover, EX = E conv(X) even if X is deterministic) where conv(X) is the convex hull of X , see [3, 25]. For instance, E f0; 1g = 0; 1] Alternatively, it is possible to define the Aumann expectation through the support function: h(X; u) sup fhu; xi: x 2 Xg ; of X , where u runs over the unit sphere S d Gamma1 , and hu; xi is the inner product of u with x. Then EX can be defined as the ....

R.J. Aumann. Integrals of set-valued functions. J. Math. Anal. Appl., 12:1--12, 1965.

....theorems. 1 Introduction The study of measurable multifunctions has been developed extensively with applications to mathematical economics and optimal control theory by many authors. The natural approach, which derives from the study of the integro di#erential inclusions, is due to Aumann in 1965 ([2]) and is # Lavoro svolto nell ambito del G.N.A.F.A. del C.N.R. e mail:matears1 unipg.it 1 based on the integration of measurable selections. Unfortunately the Aumann integral does not satisfy all the usual properties of an integral. So it seems to be natural to investigate whether the ....

R. J. AUMANN "Integrals of set valued functions", J. Math. Anal. Appl. 12, (1965) 1-12.

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R. J. Aumann. Integrals of set-valued functions. J. Math. Anal. Appl., 12:1--12, 1965.

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