Reconstruction of the One-Dimensional Lebesgue Measure

SummaryIn the Mizar system ([1], [2]), Józef Białas has already given the one-dimensional Lebesgue measure [4]. However, the measure introduced by Białas limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure to a semialgebra of intervals. Furthermore, by repeating the extension of the previous measure, we reconstructed the one-dimensional Lebesgue measure [7], [3].

Kolmogorov’s Axiomatization and Its Discontents

Kolmogorov's axiomatization of probability is the standard probability axiom system that most people learn in high school or university. And it is widely considered to be undeniably true—in much the same way that arithmetic seems to be undeniably true. However, over the years, various philosophers, mathematicians, statisticians, and scientists have found potential faults with Kolmogorov's axiom system. In this chapter these potential faults are reviewed and discussed. Among them are the problematic countable additivity, as well as finite additivity. The chapter also includes critical examinations of conditional probability, positive real numbers and sets and algebras as deployed by Kolmogorov.

Extensions of closure spaces

<p>A closure space X is a set endowed with a closure operator P(X) → P(X), satisfying the usual topological axioms, except finite additivity. A T₁ closure extension Y of a closure space X induces a structure ϒ on X satisfying the smallness axioms introduced by H. Herrlich [?], except the one on finite unions of collections. We'll use the word seminearness for a smallness structure of this type, i.e. satisfying the conditions (S1),(S2),(S3) and (S5) from [?]. In this paper we show that every T₁ seminearness structure ϒ on X can in fact be induced by a T₁ closure extension. This result is quite different from its topological counterpart which was treated by S.A. Naimpally and J.H.M. Whitfield in [?]. Also in the topological setting the existence of (strict) extensions satisfying higher separation conditions such as T₂ and T₃ has been completely characterized by means of concreteness, separatedness and regularity [?]. In the closure setting these conditions will appear to be too weak to ensure the existence of suitable (strict) extensions. In this paper we introduce stronger alternatives in order to present internal characterizations of the existence of (strict) T₂ or strict regular closure extensions.</p>