منابع مشابه
On completely nonmeasurable unions
Assume that there is no quasi-measurable cardinal not greater than 2. We show that for a c.c.c. σ-ideal I with a Borel base of subsets of an uncountable Polish space, if A is a point-finite family of subsets from I then there is a subfamily of A whose union is completely nonmeasurable i.e. its intersection with every nonsmall Borel set does not belong to the σ-field generated by Borel sets and ...
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We show that under some conditions on a family A ⊂ I there exists a subfamily A0 ⊂ A such that ⋃ A0 is nonmeasurable with respect to a fixed ideal I with Borel base of a fixed uncountable Polish space. Our result applies to the classical ideal of null subsets of the real line and to the ideal of first category subsets of the real line.
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Our main inspiration is the work in paper [4]. We will prove that for a partition A of the real line into meager sets and for any sequence An of subsets of A one can find a sequence Bn such that Bn’s are pairwise disjoint and have the same “outer measure with respect to the ideal of meager sets”. We get also generalization of this result to a class of σ-ideals posessing Suslin property. However...
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In most real analysis textbooks, the standard example of a nonmeasurable set is a subset of the real line that is due to Vitali [3]. We describe a similar nonmeasurable subset of the torus (and hence the plane), where we can more easily visualize the set. In the process of constructing the set, students get an opportunity to experience how topics they learned in algebra and topology can be used...
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This paper deals with stability of stochastic optimization problems in a general setting. Objective function is defined on a metric space and depends on a probability measure which is unknown, but, estimated from empirical observations. We try to derive stability results without precise knowledge of problem structure and without measurability assumption. Moreover, ε-optimal solutions are consid...
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ژورنال
عنوان ژورنال: Open Mathematics
سال: 2010
ISSN: 2391-5455
DOI: 10.2478/s11533-010-0038-z