تقرير
Properties of functions on a bounded charge space
| العنوان: | Properties of functions on a bounded charge space |
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| المؤلفون: | Keith, Jonathan M. |
| سنة النشر: | 2021 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Functional Analysis, 28A25, 28A20, 46E30 |
| الوصف: | A charge space $(X,\mathcal{A},\mu)$ is a generalisation of a measure space, consisting of a sample space $X$, a field of subsets $\mathcal{A}$ and a finitely additive measure $\mu$, also known as a charge. Key properties a real-valued function on $X$ may possess include $T_1$-measurability and integrability. These properties are generalisations of corresponding properties of real-valued functions on a (countably additive) measure space. However, these properties are less well studied than their measure-theoretic counterparts. This paper describes new characterisations of $T_1$-measurability and integrability in the case that the charge space is bounded, that is, $\mu(X) < \infty$. These characterisations are convenient for analytic purposes; for example, they facilitate simple proofs that $T_1$-measurability is equivalent to conventional measurability and integrability is equivalent to Lebesgue integrability, if $(X,\mathcal{A},\mu)$ is a complete measure space. Several additional contributions to the theory of bounded charges are also presented. 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 $L_1(X,\mathcal{A},\mu)$ 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 $\mathcal{L}_p(X,\mathcal{A},\mu)$. Comment: arXiv admin note: substantial text overlap with arXiv:2104.08705 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2106.10894 |
| رقم الأكسشن: | edsarx.2106.10894 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |