FINITE ADDITIVITY AND CLOSEST APPROXIMATIONS

1994 ◽  
Vol 20 (2) ◽  
pp. 559
Author(s):  
Appling
Keyword(s):  
Finite Additivity

scholarly journals Extensions of closure spaces

2003 ◽  
Vol 4 (2) ◽  
pp. 223
Author(s):  
D. Deses ◽  
A. De Groot-Van der Voorde ◽  
E. Lowen-Colebunders
Keyword(s):  
Closure Operator ◽  
Finite Additivity ◽  
Closure Space ◽  
Closure Spaces ◽  
The One ◽  
Separation Conditions

<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 T<sub>1</sub> 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 T<sub>1</sub> seminearness structure ϒ on X can in fact be induced by a T<sub>1</sub> 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 T<sub>2</sub> and T<sub>3</sub> 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) T<sub>2</sub> or strict regular closure extensions.</p>

scholarly journals Reconstruction of the One-Dimensional Lebesgue Measure

2020 ◽  
Vol 28 (1) ◽  
pp. 93-104
Author(s):  
Noboru Endou
Keyword(s):  
Lebesgue Measure ◽  
Real Space ◽  
Outer Measure ◽  
Finite Additivity ◽  
One Dimensional ◽  
The Real ◽  
Measurable Set ◽  
The One ◽  
System 1 ◽  
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].

A Conflict between Finite Additivity and Avoiding Dutch Book

1983 ◽  
Vol 50 (3) ◽  
pp. 398-412 ◽  
Author(s):  
Teddy Seidenfeld ◽  
Mark J. Schervish
Keyword(s):  
Dutch Book ◽  
Finite Additivity

Note on Continuous and Purely Finitely Additive Set Functions

1986 ◽  
Vol 29 (4) ◽  
pp. 407-412
Author(s):  
Wilfried Siebe
Keyword(s):  
Finite Additivity ◽  
Set Functions ◽  
A Charge

AbstractThe Sobczyk-Hammer respectively Yosida-Hewitt decomposition ([17], [19]) generates the class of continuous respectively purely finitely additive charges. In this paper, attention is limited to hereditable properties for these classes. It is proved that the property of continuity is preserved with respect to extensions and that if all extensions of a charge to a charge on a given field are continuous, then the original charge is continuous. An analogous heredity theorem for purely finite additivity holds true in the monogenic case.

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