On density properties of weakly absolutely continuous measures

Baklanov, A. ORCID: https://orcid.org/0000-0003-1599-3618 (2016). On density properties of weakly absolutely continuous measures. Modern Problems in Mathematics and its Applications 1662 62-72.

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Abstract

It is shown that some set of all step functions (and the set of all uniform limits of ones) allows an embedding into some compact subset (with respect to weak-star topology) of the set of all finitely additive measures of bounded variation in the form of an everywhere dense subset. Precisely, we considered the set of all step functions (the set of all uniform limits of such functions) such that integral of absolute value of the functions with respect to non-negative finitely additive measure λ is equal to the unit. For these sets, the possibility of the embedding is proved for the cases of non-atomic and finite range measure λ; in the cases the compacts do not coincide. Namely, in the nonatomic measure case, it is shown that the mentioned sets of functions allow the embedding into the unit ball (in the strong norm-variation) of weakly absolutely continuous measures with respect to λ in the form of a everywhere dense subset. In the finite range measure case, it is shown that the mentioned sets of functions allow the embedding into the unit sphere of weakly absolutely continuous measures with respect to λ in the form of a everywhere dense subset. In the last case the sphere is closed in the weak-star topology. An interpretation of these results is given in terms of an approach connected with an extension of linear control problems in the class of finitely additive measures.

Item Type:	Article
Uncontrolled Keywords:	Finitely additive measures; Non-atomic or atomless measures; Weak absolute continuity; Weak-star topology
Research Programs:	Advanced Systems Analysis (ASA)
Depositing User:	Luke Kirwan
Date Deposited:	26 Sep 2016 06:50
Last Modified:	27 Aug 2021 17:27
URI:	https://pure.iiasa.ac.at/13832

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