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This paper introduces the idea of measure over \({\alpha }-\upsigma -\) algebra and the main properties, examples, theorems, propositions and lemma for this notion has been investigated where a nonnegative extended real valued set function \(\mathcal{d}\) on \({\alpha }-\upsigma -\) algebra \(\mathfrak{J}\) is named measure over \({\alpha }-\upsigma -\) algebra \(\mathfrak{J}\) if \({\mathcal{d}}\left(\upphi \right)=0\) and \(\mathcal{d}(\bigcup_{n=1}^{\infty }{A}_{n})= \sum_{n=1}^{\infty } \mathcal{d}({A}_{n})\) whenever { \({\text{A}}_{1}{,\text{A}}_{2},\dots \) } is the family of disjoint set in \({\alpha }-\upsigma -\) algebra \(\mathfrak{J}\) . Furthermore, we study the idea of countable additive over \({\alpha }-\upsigma -\) algebra which is generalization for measure over \({\alpha }-\upsigma -\) algebra and we conclude that the notion of finite additive over \({\alpha }-\upsigma -\) algebra is weaker than of countable additive over \({\alpha }-\upsigma -\) algebra that is the countable additive over \({\alpha }-\upsigma -\) algebra implies to finite additive over \({\alpha }-\upsigma -\) algebra as well as we proved any measure over \({\alpha }-\upsigma -\) algebra is countable additive over \({\alpha }-\upsigma -\) algebra and the measure over \({\alpha }-\upsigma -\) algebra implies that to finite additive over \({\alpha }-\upsigma -\) algebra. We show that the difference of two measure over \({\alpha }-\upsigma -\) algebra is countable additive over \({\alpha }-\upsigma -\) algebra. It has been proved that a linear combination of measure over \({\alpha }-\upsigma -\) algebra is measure over \({\alpha }-\upsigma -\) algebra and the linear combination of outer measure over \({\alpha }-\upsigma -\text{algebra}\) is also outer measure over \({\alpha }-\upsigma -\) algebra. Moreover, we proved that every measure over \(\upsigma -\) algebra is measure over \({\alpha }-\upsigma -\) algebra and we concluded every outer measure over \(\upsigma -\) algebra is outer measure over \({\alpha }-\upsigma -\) algebra. Finally, it has been proved that every countable additive over \(\upsigma -\) algebra is countable additive over \({\alpha }-\upsigma -\) algebra and every finite additive over \(\upsigma -\) algebra is finite additive over \({\alpha }-\upsigma -\) algebra.

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The Properties of Types of Real Valued Set Functions Over \({\varvec{\upalpha}}-{\varvec{\upsigma}}-\) Algebra

  • 会议论文

  • 发表日期：

    2025年

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  • Hind F. Abbas,
  • Ibrahim S. Ahmed

摘要

This paper introduces the idea of measure over \({\alpha }-\upsigma -\) algebra and the main properties, examples, theorems, propositions and lemma for this notion has been investigated where a nonnegative extended real valued set function \(\mathcal{d}\) on \({\alpha }-\upsigma -\) algebra \(\mathfrak{J}\) is named measure over \({\alpha }-\upsigma -\) algebra \(\mathfrak{J}\) if \({\mathcal{d}}\left(\upphi \right)=0\) and \(\mathcal{d}(\bigcup_{n=1}^{\infty }{A}_{n})= \sum_{n=1}^{\infty } \mathcal{d}({A}_{n})\) whenever { \({\text{A}}_{1}{,\text{A}}_{2},\dots \) } is the family of disjoint set in \({\alpha }-\upsigma -\) algebra \(\mathfrak{J}\) . Furthermore, we study the idea of countable additive over \({\alpha }-\upsigma -\) algebra which is generalization for measure over \({\alpha }-\upsigma -\) algebra and we conclude that the notion of finite additive over \({\alpha }-\upsigma -\) algebra is weaker than of countable additive over \({\alpha }-\upsigma -\) algebra that is the countable additive over \({\alpha }-\upsigma -\) algebra implies to finite additive over \({\alpha }-\upsigma -\) algebra as well as we proved any measure over \({\alpha }-\upsigma -\) algebra is countable additive over \({\alpha }-\upsigma -\) algebra and the measure over \({\alpha }-\upsigma -\) algebra implies that to finite additive over \({\alpha }-\upsigma -\) algebra. We show that the difference of two measure over \({\alpha }-\upsigma -\) algebra is countable additive over \({\alpha }-\upsigma -\) algebra. It has been proved that a linear combination of measure over \({\alpha }-\upsigma -\) algebra is measure over \({\alpha }-\upsigma -\) algebra and the linear combination of outer measure over \({\alpha }-\upsigma -\text{algebra}\) is also outer measure over \({\alpha }-\upsigma -\) algebra. Moreover, we proved that every measure over \(\upsigma -\) algebra is measure over \({\alpha }-\upsigma -\) algebra and we concluded every outer measure over \(\upsigma -\) algebra is outer measure over \({\alpha }-\upsigma -\) algebra. Finally, it has been proved that every countable additive over \(\upsigma -\) algebra is countable additive over \({\alpha }-\upsigma -\) algebra and every finite additive over \(\upsigma -\) algebra is finite additive over \({\alpha }-\upsigma -\) algebra.