论文标题
在Levi-Civita Field $ \ Mathcal {R} $上的新措施上
On a new measure on the Levi-Civita field $\mathcal{R}$
论文作者
论文摘要
Levi-Civita字段$ \ Mathcal {r} $是最小的非Archimidean Ordered Field Extension的实数，而实际关闭，而Cauchy则在订单引起的拓扑中完成。在较早的论文[Shamseddine-Berz-2003]中，根据$ \ Mathcal {r} $在$ \ Mathcal {r} $上定义了一个措施，该措施是根据内部和外部和外部总和更接近的内部和外部封面长度的限制。该定义被证明在开发$ \ Mathcal {r} $上的集成理论中很有用，其中积分满足了真实分析的Lebesgue积分的许多基本属性。然而，该衡量理论缺乏一些直观的结果，人们在任何合理的措施中都会期望这一结果。例如，在另一个可测量集中的可测量集的补充不必可测量。 在本文中，我们将为[Shamseddine-Berz-2003]中定义的可测量集给出一个表征。然后，我们将在$ \ Mathcal {r} $上介绍外部度量的概念，并显示外部度量具有的一些关键属性。最后，我们将使用外部度量的概念来定义$ \ Mathcal {r} $的新措施，事实证明，这是对Lebesgue量度的更好概括，从$ \ Mathbb {r} $到$ \ Mathcal {r {r} $，并导致了一个$ \ nathcal {r} $的一系列可测量集合的家族。 [Shamseddine-berz-2003]，对于Lebesgue可测量的大多数经典结果都以$ \ Mathbb {R} $ Hold。
The Levi-Civita field $\mathcal{R}$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [Shamseddine-Berz-2003], a measure was defined on $\mathcal{R}$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over $\mathcal{R}$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [Shamseddine-Berz-2003]. Then we will introduce the notion of an outer measure on $\mathcal{R}$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on $\mathcal{R}$ that proves to be a better generalization of the Lebesgue measure from $\mathbb{R}$ to $\mathcal{R}$ and that leads to a family of measurable sets in $\mathcal{R}$ that strictly contains the family of measurable sets from [Shamseddine-Berz-2003], and for which most of the classic results for Lebesgue measurable sets in $\mathbb{R}$ hold.
