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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.13626 |
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| _version_ | 1866917410457518080 |
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| author | Behmanush, H. S. Küçükaslan, M. |
| author_facet | Behmanush, H. S. Küçükaslan, M. |
| contents | In this paper, we introduce the notion of a $γ$-density point for Lebesgue-measurable subsets of $\mathbb{R}$, where $γ$ is a modulus function, and study its basic measure-theoretic properties. We show that every $γ$-density point is a Lebesgue density point, while under Condition~(A) the two notions coincide. Consequently, for such modulus functions, the set of $γ$-density points of a measurable set differs from the set itself only by a null set, yielding a modulus version of the Lebesgue Density Theorem. We then define the associated $γ$-density topology $τ_γ$ and investigate its structure. In general, $τ_γ$ is contained in the classical Lebesgue density topology, and if $γ$ satisfies Condition~(A), then $τ_γ=τ_d$. We also compare $τ_γ$ with $ψ$-density topologies and establish several topological properties of $τ_γ$, including that countable sets are $τ_γ$-closed and that $(\mathbb{R},τ_γ)$ is nonseparable, nonregular, and nonmetrizable. Finally, we introduce $γ$-approximately continuous functions, prove that they form a vector space, and show that the bounded class of such functions is a Banach space under the supremum norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13626 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A generalization of the Lebesgue density theorem via modulus density Behmanush, H. S. Küçükaslan, M. General Topology 54D10, 54D15, 54A05, 11B05 In this paper, we introduce the notion of a $γ$-density point for Lebesgue-measurable subsets of $\mathbb{R}$, where $γ$ is a modulus function, and study its basic measure-theoretic properties. We show that every $γ$-density point is a Lebesgue density point, while under Condition~(A) the two notions coincide. Consequently, for such modulus functions, the set of $γ$-density points of a measurable set differs from the set itself only by a null set, yielding a modulus version of the Lebesgue Density Theorem. We then define the associated $γ$-density topology $τ_γ$ and investigate its structure. In general, $τ_γ$ is contained in the classical Lebesgue density topology, and if $γ$ satisfies Condition~(A), then $τ_γ=τ_d$. We also compare $τ_γ$ with $ψ$-density topologies and establish several topological properties of $τ_γ$, including that countable sets are $τ_γ$-closed and that $(\mathbb{R},τ_γ)$ is nonseparable, nonregular, and nonmetrizable. Finally, we introduce $γ$-approximately continuous functions, prove that they form a vector space, and show that the bounded class of such functions is a Banach space under the supremum norm. |
| title | A generalization of the Lebesgue density theorem via modulus density |
| topic | General Topology 54D10, 54D15, 54A05, 11B05 |
| url | https://arxiv.org/abs/2604.13626 |