Properties of Functions on a Bounded Charge Space

IF 0.9 3区数学 Q2 MATHEMATICS

Analysis and Geometry in Metric Spaces Pub Date : 2021-06-21 DOI:10.1515/agms-2022-0134

J. Keith

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引用次数: 3

Abstract

Abstract A charge space (X, 𝒜, µ) is a generalisation of a measure space, consisting of a sample space X, a field of subsets 𝒜 and a finitely additive measure µ, also known as a charge. Properties a real-valued function on X may possess include T1-measurability and integrability. However, these properties are less well studied than their measure-theoretic counterparts. This paper describes new characterisations of T1-measurability and integrability for a bounded charge space (µ(X) < ∞). These characterisations are convenient for analytic purposes; for example, they facilitate simple proofs that T1-measurability is equivalent to conventional measurability and integrability is equivalent to Lebesgue integrability, if (X, 𝒜, µ) is a complete measure space. New characterisations of equality almost everywhere of two real-valued functions on a bounded charge space are provided. Necessary and sufficient conditions for the function space L1(X, 𝒜, µ) to be a Banach space are determined. Lastly, the concept of completion of a measure space is generalised for charge spaces, and it is shown that under certain conditions, completion of a charge space adds no new equivalence classes to the quotient space ℒp(X, 𝒜, µ).

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有界电荷空间上函数的性质

摘要A电荷空间（X，𝒜, µ）是测度空间的推广，由样本空间X，子集的域组成𝒜 和有限加性测度µ，也称为电荷。X上的实值函数可能具有的性质包括T1可测性和可积性。然而，这些性质的研究不如它们的度量理论对应物深入。本文描述了有界电荷空间（µ（X）<∞）的T1可测性和可积性的新性质。这些特征便于分析；例如它们促进了T1可测性等价于常规可测性和可积性等价于Lebesgue可积性的简单证明，𝒜, µ）是一个完整的测度空间。给出了有界电荷空间上两个实值函数几乎处处相等的新性质。函数空间L1（X，𝒜, µ）是Banach空间。最后，对电荷空间推广了测度空间完备的概念，证明了在一定条件下，电荷空间完备不会给商空间增加新的等价类ℒp（X，𝒜, µ）。

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来源期刊

Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

16 weeks

期刊介绍： Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.