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1 Pure and Applied Mathematics Feb. 212 Vol. 28 No. 1 P bkc (c[, 1]) P bkc (L p [, 1]) (1) ( (), 364) (G, β, u),,, P bkc (c[, 1]) P bkc (L p [, 1]),. ; ; O A (212) , [2]. 1965, Aumann. R. J.,, 1964, Vind. K.. 197, Debreu. G. Radon-Nikodym. 1973, KendaII D. G.,. 198,, 1996, [3],. [2] (1), L[, 1]L 2 [, 1] C[, 1]. 2 P bkc (X) [3]. 1 β G.X Banach, F : β X {F (A) A β X. {sh s R s R. sh : β β, s R, ε >, λ >, A β, s s < λ, F (A + sh) F (A + s H) < ε. π A (s) = co{f (A ) A A + sh, A β, s R, A β, π A (s) X, s R, ε >, λ >, A β, s s < λ, δ(π A (s), π A (s )) ε. {F (A) A β X, [4-5],, π A (s) P bkc (X). A A + s H, (194-),,.

2 1 28 A β, A s H A, A s H β, A s H + sh A + sh, A s H + sh β. ε >, λ >, A β, s s < λ, F (A + sh) F (A + s H) < ε, F (A ) F (A s H + sh) = F ((A s H) + s H) F ((A s H) + sh) < ε. π 1 A (s) = {F (A ) A A + sh, A β d(f (A ), π A (s)) d(f (A ), π 1 A(s)) d(f (A ), F (A s H + sh)) < ε, A A + s H, A β, x π 1 A (s ), d(x, π A (s)) < ε, [5] co B = {B, x co π 1 A (s ), n {x 1, x 2,, x n πa(s 1 ), α i, α i = 1, x = n α i x i. d(x i, π A (s)) < ε), y i π A (s), x i y i < ε, i = 1, 2,, n, y = n α i y i π A (s), d(x, y) n α i x i y i < ε, d(x, π A (s)) d(x, y) < ε, x co π 1 A (s ), sup d(x, π A (s)) ε. x co πa 1 (s ) x π A (s ), {x n co π 1 A (s ), x n x, d(x, π A (s)) x, d(x, π A (s)) ε, sup d(x, π A (s)) ε. x π A (s ) sup d(x, π A (s )) ε. x π A (s) A β, s s < λ, δ(π A (s), π A (s )) ε. 1 [3] (G, β), X, π : G P f (X), U X, π 1 (U) = {s G π(s) U β.. 2 (G, β), X Banach, F : β X {F (A) A β X, {sh s R s R, sh : β β, s R, ε >, λ >, A β, s s < λ, F (A + sh) F (A + s H) < ε, π A (s) = co{f (A ) A A + sh, A β, s R, A β,

3 1 : P bkc (c[, 1]) P bkc (L p [, 1]) (1) 11 π A (s) P bkc (X). 1 π A (s) P bkc (X), 1 [3] 1.2.5, ε >, λ > x X, A β, s s < λ, d(x, π A (s)) d(x, π A (s )) ε, x X, A β, d(x, π A (s)) s, d(x, π A (s)) (R, B(R)), B(R) R Borel σ, X, [3] π A (s). 1 2 : 1 (G, β),x Banach, F : β X, {sh s R s R, sh : β β, s R, ε >, λ >, A β, s s < λ, F (A + sh) F (A + s H) < ε. π A (s) = co{f (A ) A A + sh, A β, s R, A β, s R, ε >, λ > A β, s s < λ, δ(π A (s), π A (s )) ε, π A (s) P fc (X). 3 (G, β, µ) θ, H, h θ, A β, πa(s) 1 = co{µ(a + th) A A + sh, A β, { x πa(s) 2 = co µ(a + th)dt A A + sh, A β, { πa(s) 3 = co k(x, t)µ(a + th)dt A A + sh, A β, π i A (s) P bkc(c[, 1]), k(x, t) [, 1] [, 1], i = 1, 2, 3, s R. F 1 (A) = µ(a+th), A β, [6] 4, h θ, {µ(a+th) A β, t R, ε >, λ >, A β, t 2 t 1 < λ, µ(a + t 2 h) µ(a + t 1 h) < ε, F 1 (A) C[, 1], {F 1 (A) A β C[, 1], s R, ε >, λ >, A β, s s < λ, F 1 (A + sh) F 1 (A + s H) c[,1] = µ(a + th + sh) µ(a + th + s H) c[,1] < ε, c[, 1] Banach, 2 π 1 A (s) P bkc(c[, 1]). F 2 (A) c[, 1], F 2 (A) = x { x {F 2 (A) A β = µ(a + th)dt, A β, x 1, µ(a + th)dt A β

4 12 28, {F 2 (A) A β c[, 1], s R, [6] 4, ε >, λ >, A β, s s < λ, F 2 (A + sh) F 2 (A + s H) c[,1] x x = µ(a + th + sh)dt µ(a + th + s H)dt µ((a + th) + sh) µ((a + th) + s H) dt < ε. c[, 1], 2 π 2 A (s) P bkc(c[, 1]). F 3 (A) = K(x, t), f(t) c[, 1], (Kf)(x) = K(x, t)µ(a + th)dt, A β, K(x, t)f(t)dt, [4] K c[, 1] c[, 1], F 3 (A) c[, 1], {F 3 (A) A β c[, 1], K(x, t) [, 1] [, 1],, K(x, t) L, [6] 4, ε >, λ >, A β, s s < λ, µ(a + sh) µ(a + s H) < ε L. F 3 (A + sh) F 3 (A + s H) c[,1] = K(x, t)µ(a + th + sh)dt K(x 1, t)µ(a + th + s H)dt K(x, t) µ((a + th) + sh) µ((a + th) + s H) dt < ε. c[, 1], 2 π 3 A (s) P bkc(c[, 1]). 4 (G, β, µ) θ, H, h θ, A β, πa(s) 4 = co{µ(a + th) A A + sh, A β, { x πa(s) 5 = co µ(a + th)dt A A + sh, A β, { πa(s) 6 = co k(x, t)µ(a + th)dt A A + sh, A β { πa(s) 7 = co k(x, t)µ(a + th)dt A A + sh, A β k(x, t) [, 1] [, 1], k(x, t) L 2 ([, 1] [, 1]) s R, π i A (s) P bkc (L P [, 1]) P 1, i = 4, 5, 6, π 7 A (s) P bkc(l 2 [, 1]).,,

5 1 : P bkc (c[, 1]) P bkc (L p [, 1]) (1) 13 F 4 (A) = µ(a + th), A β, F 5 (A) = x µ(a + th)dt, A β, y 1 (t), y 2 (t) c[, 1], F 6 (A) = x k(x, t)µ(a + th)dt, A β. y 1 (t) y 2 (t) L P [,1] y 1 (t) y 2 (t) c[,1]. 3 {F 1 (A) A β, {F 2 (A) A β {F 3 (A) A β c[, 1], {F 4 (A) A β, {F 5 (A) A β {F 6 (A) A β L P [, 1], 3 F 4 (A + sh) F 4 (A + s ) L P [,1] F 1 (A + sh) F 1 (A + s H) c[,1], F 5 (A + sh) F 5 (A + s ) L P [,1] F 2 (A + sh) F 2 (A + s H) c[,1], F 6 (A + sh) F 6 (A + s ) L P [,1] F 3 (A + sh) F 3 (A + s H) c[,1] s R, ε >, λ >, A β, s s < λ, F i (A + sh) F i (A + s H) L P [,1] ε, i = 4, 5, 6, L P [, 1], 2, π i A (s) P bkc(l P [, 1]), i = 4, 5, 6. F 7 (A) = k(x, t)µ(a + th)dt, A β, k(x, t), k(x, t) L, f(t) L 2 [, 1], (kf)(x) = k(x, t)f(t)dt, [4] k L 2 [, 1] L 2 [, 1], F 7 (A) L 2 [, 1], {F 7 (A) A β, L 2 [, 1]. [6] 4, s R, ε >, λ >, A β, s s < λ, µ(a + sh) µ(a + s H) < ε L, F 7 (A + sh) F 7 (A + s ) L 2 [,1] k(x, t) µ((a + th) + sh) µ((a + th) + s H) dt ε. L 2 [, 1], 2, π 7 A (s) P bkc(l 2 [, 1]). 3 2 [7] XY, π : X P (Y ), x X, π x (usc), π(x ) U, x V, x V, π(x) U. π x X usc, π (usc).

6 14 28 π x (lsc), U, π(x ) U, x V, x V, π(x) U. π x X lsc, π (lsc). π x, π x. π, π. 3 [7] X, Y, π : X P (Y ), x X π x Hausdorff (Hlsc), ε >, x U, π(x ) x U {y d(y, π(x)) < ε. X Hausdorff, Y Hausdorff, y Y, x σ(y, π(x)) x, π x h- (husc), σ(y, π(x)) π. 5 β G, X Banach, F : β X {F (A) A β X. {sh s R s R, sh : β β, s R, ε >, λ >, A β, s s < λ, F (A + sh) F (A + s H) < ε. π A (s) = co{f (A ) A A + sh, A β, s R, A β, A β, π A (s) P bkc (X) h-. 1 π A (s) P bkc (X), s R, ε >, λ >, A β, s s < λ, δ(π A (s), π A (s )) ε 2. [3] 1.2.5, π A (s ) {x d(x, π A (s)) < ε, s (s λ, s + λ), 3 π A s R Hausdorff (Hlsc). [3] , A β, s s < λ, δ(π A (s), π A (s )) = sup σ(x, π A (s)) σ(x, π A (s )) ε x 1 2. : a, σ(ax, B) = aσ(x, B), x X, σ(x, π A (s)) s R, σ(x, π A (s)) s R, 3 π A s R h- (husc), π A s R Hausdorff (Hlsc), s R h- (husc). π A (s) h-. [7] , π A (s). 6 (G, β, µ) θ, H, h θ, A β, 3 π 1 A (s), π2 A (s), π3 A (s) P bkc(c[, 1]); 4 π 4 A (s), π5 A (s), π6 A (s) P bkc (L P [, 1]); π 7 A (s) P bkc(l 2 [, 1]) h {F i (A) A β,, s R, ε >, A β, s s < λ, F i (A + sh) F i (A + s H) < ε. i = 1, 2, 3, 4, 5, 6, 7,

7 1 : P bkc (c[, 1]) P bkc (L p [, 1]) (1) β G, X Banach, F : β X {F (A) A β X. Q s Q, s : β β, s Q, A β, ε >, s V, A s A, A β, s V, F (A ) F (A s + s ) < ε. π A (s) = co{f (A ) A A + s, A β, s Q, π S (s) P bkc (X) s (usc), π A (s) s h- (husc), s V, sup d(x, π A (s )) ε. x π A (s) 1 π A (s) X, A A + s, A β, A s + s A + s, A s + s β. π 1 A(s) = {F (A ) A A + s, A β, s Q, A s A, A β, s V, F (A ) F (A s + s ) < ε. d(f (A ), π A (s )) d(f (A ), π 1 A(s )) d(f (A ), F (A s + s )) < ε, A A + s, A β, x π 1 A (s), d(x, π A(s )) < ε [5], co B = {B, x co π 1 A (s), 1 d(x, π A(s )) < ε, x co πa 1 (s), 1, sup d(x, π A (s )) ε, s V. x co πa 1 (s) sup d(x, π A (s )) ε, s V. x π A (s) U π A (s ), U U C, π A (s ) U C =, x 1 π A (s ), ( B x 1, 1 ) { 2 d(x1, U C ) = x d(x, x 1 ) < 1 2 d(x1, U C ) U, { ( B x 1, 1 2 d(x1, U )) C x 1 π A (s ) π A (s ), {x 1, x 2,, x n π A (s ) π A (s ) n ( B x i, 1 ) 2 d(x i, U C ) U.

8 16 28 { 1 2ε = min 2 d(x 1, U C ),, 1 2 d(x n, U C ), x d(x, π A (s )) < 2ε, x π A (s ), d(x, x) < 2ε, π A (s ) n j, x B(x j, 1 2 d(x j, U C )), j n, ( B x i, 1 ) 2 d(x i, U C ) d(x, x j ) d(x, x) + d(x, x j ) < 2ε d(x j.u C ) d(x j, U C ), x U, {x d(x, π A (s )) < 2ε U. s V, sup d(x, π A (s )) ε, s V, x π A (s) π A (s) {x d(x, π A (s )) ε {x d(x, π A (s )) < 2ε U, 2 π A (s) s (usc). x X, ε >, { π A (s ) x d(x, π A(s )) < ε x, π A (s) s (usc), s V, s V, { π A (s) x d(x, π A(s )) < ε { x = π A (s ) + x x <, s V, ( { σ(x, π A (s)) σ(x, π A (s )) + σ x, x x < ε x. ε ) x < σ(x, π A (s )) + ε. x X, σ(x, π A (s)) s, 3 π A (s) s h- (husc). 2 β G, X Banach, F : β X {F (A) A β X. Q s Q, s : β β, A β, s Q, ε >, s V, A s A, A β, s V, F (A ) F (A s + s ) < ε. π A (s) = co{f (A ) A A + s, A β, s Q, π A (s) P bkc (x) Q (usc), (π A ).

9 1 : P bkc (c[, 1]) P bkc (L p [, 1]) (1) 17 7 [7] β G, X Banach, F : β X, Q s Q, s : β β, s Q, A β, ε >, s V, A s A, A β, s V, F (A ) F (A s + s) < ε. π A (s) = co{f (A ) A A + s, A β, s Q, π A (s) P fc (x) s Hausdorff (Hlsc), s (lsc), sup d(x, π A (s)) ε, s V. x π A (s ), π A (s) P fc (x) A A+s, A β, A s +s A + s, A s + s β. A s A, A β, s V, π 1 A(s) = {F (A ) A A + s, A β, s Q, d(f (A ), π A (s)) d(f (A ), π 1 A(s)) F (A ) F (A s + s) < ε. A A + s, A β, x π 1 A (s ), d(x, π A (s)) < ε, s V. 1 : sup d(x, π A (s)) ε, s V. x π A (s ) x π A (s ), d(x, π A (s)) ε, s V π A (s ) {x d(x, π A (s)) ε, s V. 3 π A (s) s Hausdorff (Hlsc). U, π A (s ) U, x π A (s ) U, λ >, {x d(x, x ) < λ U. π A (s) s Hausdorff (Hlsc), s V, s V, π A (s ) {x d(x, π A (s)) < λ. x π A (s ) {x d(x, π A (s)) < λ, x π A (s), d(x, x ) < λ, x U, s V,π A (s) U, π A (s) s (lsc). Y, Q Y, x Q, a >, ax Q [7]. 9 β G, X Banach, F : β X, Q s Q, s : β β, A β s Q ε >, < t < 1, A A + as, A β, a (1 t, 1), F (A ) F (A + (1 a)s ) < ε. b (1, 1 + t), A + bs A + (2 b)s, π A (s) P fc (X) a 1 < t, π A (s) = co{f (A ) A A + s, A β, s Q, sup d(x, π A (s )) ε. x π A (as )

10 18 28 π A (s) P fc (X), a (1 t, 1), A A + as, A β A + (1 a)s A + as + (1 a)s = A + s, A + (1 a)s β. π 1 A(s) = {F (A ) A A + s, A β, s Q, A A + as, A β, a (1 t, 1), F (A ) F (A + (1 a)s ) < ε, d(f (A, π A (s )) d(f (A ), π 1 A(s )) d(f (A ), F (A + (1 a)s )) < ε. A A + as, A β, x π 1 A (as ), d(x, π A (s )) < ε, a (1 t, 1). b (1, 1+t), A A+bs, A β, A+bs A+(2 b)s, A A+(2 b)s, 2 b (1 t, 1), d(f (A ), π A (s )) < ε, A A + bs, A β x π 1 A (bs ), d(x, π A (s )) < ε, b (1, 1 + t). a 1 < t, x π 1 A (as ), d(x, π A (s )) < ε, 1 sup d(x, π A (s )) ε, a 1 < t. x π A (as ) [1]. [M]. :, 29. [2]. [J]., 21,26(5), [3],,. [M]. :, [4],,,. : [M]. :, 198. [5],,,. [M]. :, 29. [6]. [J]. : A, 1987,8(6): [7],. [M]. :, 24. Value of set-valued stochastic variables between P bkc (c[, 1]) and P bkc (L p [, 1]) Lin Yixing (Longyan Normal School, Longyan 364, China) Abstract: In this paper,we study the quasi-continuous measure space (G, β, µ) of uniformly boundness of continuous functions, and the containment relations of convex and closure set. We have constructed value of set-valued stochastic variables and continuous set-valued mapping between P bkc (c[, 1]) and P bkc (L p [, 1]). It will deepen the set-valued stochastic process theory. Key words: quasi-continuous measure space, set-valued stochastic variable, continuous set-valued mapping 21 MSC: 28B2

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