This paper gives internal characterizations of some sequence covering compact images and compact covering compact images of paracompact locally compact spaces, which improve some results on compact images of locally...This paper gives internal characterizations of some sequence covering compact images and compact covering compact images of paracompact locally compact spaces, which improve some results on compact images of locally compact metric spaces.展开更多
In this note, we consider the multipliers on weighted function spaces over totally disconnected locally compact abelian groups (Vilenkin groups). Firstly we show an (H1 ,L ) multiplier result. We also give an (Hap ,Ha...In this note, we consider the multipliers on weighted function spaces over totally disconnected locally compact abelian groups (Vilenkin groups). Firstly we show an (H1 ,L ) multiplier result. We also give an (Hap ,Hap) multiplier result under a similiar condition of Lu Yang type. In section 2, we obtain a result about the boundedness of multipliers on weighted Besov spaces.展开更多
This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference be...This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The motivation arises from studying random elements talking values in N.B.A. Those topological properties are important assumptions that enable us to avoid possible difficulties when generalising concepts of random variable convergence, the definition of conditional law and others. For each N.B.A., there exists a finite measure space ( E,ℰ,μ ) such that the N.B.A. is isomorphic to ( ℰ ˜ , μ ˜ ) resulting from the factorisation of initial σ-algebra by the ideal of negligible sets. We focus on topological properties ( ℰ ˜ , μ ˜ ) in general setting when μ can be an infinite measure. In case when μ is infinite, we also consider properties of ℰ ˜ fin ⊆ ℰ ˜ consisting of classes of measurable sets having finite measure. The compactness and separability of the N.B.A. are characterised using the newly defined terms of approximability and uniform approximability of the corresponding measure space. Finally, conditions on ( E,ℰ,μ ) are derived for separability and compactness of ℰ ˜ and ℰ ˜ fin .展开更多
In this paper, some results concerning the relationship between the bounded-ness of some spheres and the local boundedness of the .F*-space are presented. Moreover, some results about the compactness are also given.
Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The gen...Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement <sup>*</sup><em>X </em>of the space <em>X</em> which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every <em>f</em> in <em>C</em>(<em>X</em>,<em>R</em>) (the space of bounded continuous real valued functions on <em>X</em>) or <em>Cc</em>(<em>X</em>,<em>R</em>) (the space of continuous real valued functions on <em>X</em> with compact support) or the dual group <span style="white-space:nowrap;">Γ </span>of the locally compact Abelian group <em>G</em> is extended to the set <img alt="" src="Edit_b9535172-924d-44f0-bab3-c49db17a3b7a.png" /> of the above mentioned equivalence classes. A compact topology on <img alt="" src="Edit_9d7962a3-b8a3-4693-b62a-078c8c4b4853.png" /> is obtained as the weak topology generated by these extensions of <em>f</em>. Then <em>X</em> is naturally imbedded densely in <img alt="" src="Edit_f7d403b2-eff3-4555-b8e7-1b106e06d2e7.png" />.展开更多
文摘This paper gives internal characterizations of some sequence covering compact images and compact covering compact images of paracompact locally compact spaces, which improve some results on compact images of locally compact metric spaces.
文摘In this note, we consider the multipliers on weighted function spaces over totally disconnected locally compact abelian groups (Vilenkin groups). Firstly we show an (H1 ,L ) multiplier result. We also give an (Hap ,Hap) multiplier result under a similiar condition of Lu Yang type. In section 2, we obtain a result about the boundedness of multipliers on weighted Besov spaces.
文摘This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The motivation arises from studying random elements talking values in N.B.A. Those topological properties are important assumptions that enable us to avoid possible difficulties when generalising concepts of random variable convergence, the definition of conditional law and others. For each N.B.A., there exists a finite measure space ( E,ℰ,μ ) such that the N.B.A. is isomorphic to ( ℰ ˜ , μ ˜ ) resulting from the factorisation of initial σ-algebra by the ideal of negligible sets. We focus on topological properties ( ℰ ˜ , μ ˜ ) in general setting when μ can be an infinite measure. In case when μ is infinite, we also consider properties of ℰ ˜ fin ⊆ ℰ ˜ consisting of classes of measurable sets having finite measure. The compactness and separability of the N.B.A. are characterised using the newly defined terms of approximability and uniform approximability of the corresponding measure space. Finally, conditions on ( E,ℰ,μ ) are derived for separability and compactness of ℰ ˜ and ℰ ˜ fin .
基金This research is supported by National Natural Science Foundation of China(19971046) RFDP(2001005513)
文摘In this paper, some results concerning the relationship between the bounded-ness of some spheres and the local boundedness of the .F*-space are presented. Moreover, some results about the compactness are also given.
文摘Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement <sup>*</sup><em>X </em>of the space <em>X</em> which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every <em>f</em> in <em>C</em>(<em>X</em>,<em>R</em>) (the space of bounded continuous real valued functions on <em>X</em>) or <em>Cc</em>(<em>X</em>,<em>R</em>) (the space of continuous real valued functions on <em>X</em> with compact support) or the dual group <span style="white-space:nowrap;">Γ </span>of the locally compact Abelian group <em>G</em> is extended to the set <img alt="" src="Edit_b9535172-924d-44f0-bab3-c49db17a3b7a.png" /> of the above mentioned equivalence classes. A compact topology on <img alt="" src="Edit_9d7962a3-b8a3-4693-b62a-078c8c4b4853.png" /> is obtained as the weak topology generated by these extensions of <em>f</em>. Then <em>X</em> is naturally imbedded densely in <img alt="" src="Edit_f7d403b2-eff3-4555-b8e7-1b106e06d2e7.png" />.
