scholarly journals Filters and linked families of sets

2020 ◽  
Vol 30 (3) ◽  
pp. 444-467
Author(s):  
A.G. Chentsov
Keyword(s):  
Measurable Space ◽  
The Family ◽  
System Product ◽  
Bitopological Spaces

Properties of ultrafilters (u/f) and maximal linked systems (MLS) on the widely understood measurable space (MS) and representations of linked (not necessarily maximal) families and filters on this MS are investigated. Conditions realizing maximality of linked families (systems) and natural representations for bitopological spaces (BTS) of u/f and MLS are established. Equipments of sets of linked families and filters corresponding to Wallman and Stone schemes are studied; the connection of these equipments with analogous equipments (with topologies) for u/f and MLS leading to above-mentioned BTS is studied too. Properties of linked family products for two (widely understood) MS are investigated. It is shown that MLS on the π-system product (that is, on the family of “measurable” rectangles) are limited to products of corresponding MLS on initial spaces.

scholarly journals On certain analogues of linkedness and supercompactness

2020 ◽  
Vol 55 ◽  
pp. 113-134
Author(s):  
A.G. Chentsov
Keyword(s):  
Topological Space ◽  
The Family ◽  
Measurable Structure ◽  
Bitopological Spaces ◽  
Algebra Of Sets

Natural generalizations of properties of the family linkedness and the topological space supercompactness are considered. We keep in mind reinforced linkedness when nonemptyness of intersection of preassigned number of sets from a family is postulated. Analogously, supercompactness is modified: it is postulated the existence of an open subbasis for which, from every covering (by sets of the subbasis), it is possible to extract a subcovering with a given number of sets (more precisely, not more than a given number). It is clear that among all families having the reinforced linkedness, one can distinguish families that are maximal in ordering by inclusion. Under natural and (essentially) “minimal”' conditions imposed on the original measurable structure, among the mentioned maximal families with reinforced linkedness, ultrafilters are certainly contained. These ultrafilters form subspaces in the sense of natural topologies corresponding conceptually to schemes of Wallman and Stone. In addition, maximal families with reinforced linkedness, when applying topology of the Wallman type, have the above-mentioned property generalizing supercompactness. Thus, an analogue of the superextension of the $T_1$-space is realized. The comparability of “Wallman”' and “Stone”' topologies is established. As a result, bitopological spaces (BTS) are realized; for these BTS, under equipping with analogous topologies, ultrafilter sets are subspaces. It is indicated that some cases exist when the above-mentioned BTS is nondegenerate in the sense of the distinction for forming topologies. At that time, in the case of “usual” linkedness (this is a particular case of reinforced linkedness), very general classes of spaces are known for which the mentioned BTS are degenerate (the cases when origial set, i.e., “unit”' is equipped with an algebra of sets or a topology).

scholarly journals The Radon-Nikodym Theorem for Multimeasures

1978 ◽  
Vol 21 (2) ◽  
pp. 167-173
Author(s):  
Le van Tu
Keyword(s):  
Convex Space ◽  
Measurable Space ◽  
Locally Convex Space ◽  
The Family ◽  
Disjoint Sets ◽  
Locally Convex ◽  
Nikodym Theorem

Let (S, ℳ) be ameasurable space(that is, a setSin which is defined a σ-algebra ℳ of subsets) andXa locally convex space. A mapMfrom ℳ to the family of all non-empty subsets ofXis called a multimeasure iff for every sequence of disjoint setsAnɛ ℳ (n=1,2,… )withthe seriesconverges (in the sense of (6), p. 3) toM(A).

Maximal linked systems and ultrafilters: main representations and topological properties

2020 ◽  
pp. 68-84
Author(s):  
Aleksandr G. Chentsov
Keyword(s):  
Measurable Space ◽  
Topological Properties ◽  
Bitopological Space ◽  
Measurable Structure ◽  
Bitopological Spaces ◽  
Algebras Of Sets

Questions connected with representation of the ultrafilter (UF) set for widely understood measurable space are investigated; this set is considered as a subspace of bitopological space of maximal linked systems (MLS) under equipment with topologies of Wallman and Stone types (measurable structure is defined as a π -system with “zero” and “unit”). Analogous representations connected with generalized variant of cohesion is considered also; in this variant, for corresponding set family, it is postulated the nonemptyness of intersection for finite subfamilies with power not exceeding given. Conditions of identification of UF and MLS (in the above-mentioned generalized sense) are investigated. Constructions reducing to bitopological spaces with points in the form of MLS and 𝑛-supercompactness property generalizing the “usual” supercompactness are considered. Finally, some characteristic properties of MLS and their corollaries connected with the MLS contraction to a smaller π -system are being studied. The case of algebras of sets is selected separately.

MAXIMAL LINKED SYSTEMS AND ULTRAFILTERS OF WIDELY UNDERSTOOD MEASURABLE SPACES

2018 ◽  
pp. 846-860
Author(s):  
Aleksandr Georgievich Chentsov
Keyword(s):  
Sufficient Conditions ◽  
Measurable Space ◽  
Necessary And Sufficient Conditions ◽  
Bitopological Space ◽  
Necessary And Sufficient ◽  
Bitopological Spaces

Two types of set families (ultrafilters or maximal filters and maximal linked systems) for widely understood measurable space are considered. The resulting sets of ultrafilters and maximal linked systems are equipped with the pair of comparable topologies (within the meaning of «Wallman» and «Stone»). As a result, two bitopological spaces are realized; one of them turns out a subspace of another. More precisely, ultrafilters are maximal linked systems and the totality of the latter forms a cumulative bitopological space. With employment of topological constructions some characteristic properties of ultrafilters and (in smaller power) maximal linked systems are obtained (the question is necessary and sufficient conditions of maximality of filters and linked systems).

scholarly journals Some topological properties of the space of maximal linked systems with topology of Wallman type

2020 ◽  
Vol 56 ◽  
pp. 122-137
Author(s):  
A.G. Chentsov
Keyword(s):  
Measurable Space ◽  
Topological Properties ◽  
Bitopological Spaces ◽  
Space Conditions ◽  
Degenerate Cases

Maximal linked systems (MLS) and ultrafilters (u/f) on a widely understood measurable space (this is a nonempty set with equipment in the form of π-system with “zero” and “unit”) are investigated. Under equipment with topology of Wallman type, the set of MLS is converted into a supercompact T1-space. Conditions under which given space of MLS is a supercompactum (i.e., a supercompact T2-space) are investigated. These conditions then apply to the space of u/f under equipment with topology of Wallman type. The obtained conditions are coordinated with representations obtained under degenerate cases of bitopological spaces with topologies of Wallman and Stone types, but they are not the last to be exhausted.

scholarly journals Random fixed point theorems for multivalued nonexpansive non-self-random operators

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
S. Plubtieng ◽  
P. Kumam
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Compact Convex ◽  
Measurable Space ◽  
Contractive Mapping ◽  
Random Operator ◽  
Random Operators ◽  
Random Fixed Point ◽  
The Family ◽  
Separable Subset

Let (Ω,Σ) be a measurable space, with Σ a sigma-algebra of subset of Ω, and let C be a nonempty bounded closed convex separable subset of a Banach space X, whose characteristic of noncompact convexity is less than 1, KC(X) the family of all compact convex subsets of X. We prove that a multivalued nonexpansive non-self-random operator T:Ω×C→KC(X), 1-χ-contractive mapping, satisfying an inwardness condition has a random fixed point.

Triassic sponges (Sphinctozoa) from Hells Canyon, Oregon

1988 ◽  
Vol 62 (03) ◽  
pp. 419-423 ◽  
Author(s):  
Baba Senowbari-Daryan ◽  
George D. Stanley
Keyword(s):  
Endemic Species ◽  
Upper Triassic ◽  
Island Arc ◽  
Tropical Island ◽  
The Family ◽  
Wallowa Terrane ◽  
Unique Condition ◽  
Hells Canyon

Two Upper Triassic sphinctozoan sponges of the family Sebargasiidae were recovered from silicified residues collected in Hells Canyon, Oregon. These sponges areAmblysiphonellacf.A. steinmanni(Haas), known from the Tethys region, andColospongia whalenin. sp., an endemic species. The latter sponge was placed in the superfamily Porata by Seilacher (1962). The presence of well-preserved cribrate plates in this sponge, in addition to pores of the chamber walls, is a unique condition never before reported in any porate sphinctozoans. Aporate counterparts known primarily from the Triassic Alps have similar cribrate plates but lack the pores in the chamber walls. The sponges from Hells Canyon are associated with abundant bivalves and corals of marked Tethyan affinities and come from a displaced terrane known as the Wallowa Terrane. It was a tropical island arc, suspected to have paleogeographic relationships with Wrangellia; however, these sponges have not yet been found in any other Cordilleran terrane.

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