Star-finite representations of measure spaces

Anderson, Robert M.

doi:10.1090/s0002-9947-1982-0654856-1

Cited by 77 publications

(28 citation statements)

References 16 publications

(13 reference statements)

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“…A sequence of measurable mappings from a measure space to a topological space is tight if for any Ͼ 0, there exists a compact set containing more than (1 Ϫ ) of the mass of the measure induced by each mapping. The tightness hypothesis guarantees that the standard part map (14,(16)(17)(18)24) is well-defined and Loeb measurable, thereby furnishing the measurability hypothesis in Corollary 1. We can now present, in the notation of Corollary 1, our next result.…”

Section: With (Z L(ᐆ) L()) Its Loeb Standardization; and (Iii) A Pamentioning

confidence: 82%

Nonatomic games on Loeb spaces

Khan

Sun

1996

Proc. Natl. Acad. Sci. U.S.A.

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In the setting of noncooperative game theory, strategic negligibility of individual agents, or diffuseness of information, has been modeled as a nonatomic measure space, typically the unit interval endowed with Lebesgue measure. However, recent work has shown that with uncountable action sets, for example the unit interval, there do not exist purestrategy Nash equilibria in such nonatomic games. In this brief announcement, we show that there is a perfectly satisfactory existence theory for nonatomic games provided this nonatomicity is formulated on the basis of a particular class of measure spaces, hyperfinite Loeb spaces. We also emphasize other desirable properties of games on hyperfinite Loeb spaces, and present a synthetic treatment, embracing both large games as well as those with incomplete information.

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Section: With (Z L(ᐆ) L()) Its Loeb Standardization; and (Iii) A Pamentioning

confidence: 82%

Nonatomic games on Loeb spaces

Khan

Sun

1996

Proc. Natl. Acad. Sci. U.S.A.

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“…)zC(K,,,R) and so, using Anderson's Luzin theorem [2] applied to the measurable function fk: [0, ~) ~ C(Km, R), we see that for a.a. finite z °Fdz, U) =fd°r, °U) for all U with IIUII < oo. )zC(K,,,R) and so, using Anderson's Luzin theorem [2] applied to the measurable function fk: [0, ~) ~ C(Km, R), we see that for a.a. finite z °Fdz, U) =fd°r, °U) for all U with IIUII < oo.…”

Section: Bochner Integration and Applications To Navier-stokes Equationsmentioning

confidence: 97%

“…The stochastic integral ~g dw is defined in [1][2][3][4][5][6][7][8][9][10][11][12] Note that if we write Wk(t) = (w(t), ek) and wtm~(t) = ~ w~(t)e~, k=l then w t~) is a Wiener process with covariance Q., = Pr,.Q Pr,. and fo fo…”

Section: Standard Preliminariesmentioning

confidence: 99%

“…The precise definition of the solution is given at the end of Section 2 after we have introduced the necessary notations. This solution has slightly better pathwise regularity than in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], where this case was considered. This solution has slightly better pathwise regularity than in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], where this case was considered.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic Navier-Stokes equations

Capiński

Cutland

1991

Acta Appl Math

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We construct a solution to stochastic Navier-Stokes equations in dimension n ~< 4 with the feedback in both the external forces and a general infinite-dimensional noise. The solution is unique and adapted to the Brownian filtration in the 2-dimensional case with periodic boundary conditions or, when there is no feedback in the noise, for the Dirichlet boundary condition. The paper uses the methods of nonstandard analysis.AMS subject classifications (1991). 76D05, 35Q30, 60H15, 26E35, 28E05.

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“…By Theorem 3.8 in[2] dora(G) is then #-measurable. As in the proof of Proposition 3.1 the *Y-sections of the graph st-l(G) are II~ sets.…”

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confidence: 89%

Graphs with ∏ 1 0 (K)Y-sections

Živaljević

1993

Arch Math Logic

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We prove that a Borel subset of the product of two internal sets X and Y all of whose Y-sections are II~176 sets is the intersection (union) of a countable sequence of Borel graphs with internal Y-sections. As a consequence we prove some standard results about the domains of graphs in the product of two topological spaces all of whose horizontal section are compact (open) sets. A version of classical Vitali-Lusin theorem for those types of graphs is given as well as a new proof (and an extension) of a classical result of Kunugui. IntroductionThe investigation of structure of graphs in the product X • Y of two internal sets X and Y with Y-sections of fixed Borel class (in the Descriptive Set Theory of Hyperfinite Sets) has been inspired by a curious result of the authors of [7] about the structure of Borel functions. They proved that any function whose graph is a Borel subset of the product X • Y can be covered by countably many internal functions (provided that the nonstandard universe is cvl-saturated ). This in turn shows that any Borel function can be expressed as a countable union of restrictions of internal functions to disjoint Borel sets in X. In [13] this result has been extended to arbitrary Borel graphs all of whose Y-sections are internal sets. Earlier, the Loeb measure preserving property of injective Borel functions has been noticed by the authors of [5], who in fact proved that any Borel function possesses an internal lifting (in our terminology).

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Star-finite representations of measure spaces

Cited by 77 publications

References 16 publications

Nonatomic games on Loeb spaces

Nonatomic games on Loeb spaces

Stochastic Navier-Stokes equations

Graphs with ∏ 1 0 (K)Y-sections

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