537 research outputs found
On p-Convergence in Measure of a Sequence of Measurable Functions
No full textIn the study by Papanastassiou and Papachristodoulos, 2009 the notion of p-convergence in measure was introduced. In a natural way p-convergence in measure induces an equivalence relation
on the space M of all sequences of measurable functions converging in
measure to zero. We show that the quotient space ℳ is a complete but
not compact metric space
Schur and Nikodym convergence-type theorems in Riesz spaces with respect to the (r)-convergence
No full textSome versions of Schur and Nikodym convergence-type theorems for Riesz space-valued countably additive measures with respect to the (r)-convergence are given. The concept of sigma-additive measure is formulated similarly as in the classical case (not with respect to a common unit), while the notion of pointwise convergence of the involved measures is given with respect to a same unit
Schur and matrix theorems with respect to I-convergence
No full textIn this paper we deal with the problem of proving some versions of limit theorems when the pointwise convergence of the measures involved is replaced by the weaker ideal pointwise convergence. In the general case, the answer is negative, as it is shown by means of an example. However, it is possible to do some answers, in particular cases. In this paper, as examples of limit theorems, we prove some Schur-type and basic matrix theorems with respect to ideal convergence
Ideal convergence and divergence of nets in l-groups
No full textIn this paper we introduce the I- and I^*-convergence and divergence of nets in l-groups. We prove some theorems relating
different types of convergence/divergence for nets in l-group setting, in relation with ideals. We consider both order and (D)-convergence.
By using basic properties of order sequences,
some fundamental properties, Cauchy-type characterizations
and comparison results are derived.
We prove that I^-convergence/divergence implies
I-convergence/divergence for every ideal, admissible for
the set of indexes with respect to which the net involved is directed, and we investigate a class of ideals for which the converse implication holds
Schur lemma and limit theorems in lattice groups with respect to filters
No full textSome Schur, Nikodym, Brooks-Jewett and Vitali-Hahn-Saks-type
theorems for l-group-valued measures are proved in the setting
of filter convergence. Using some properties of diagonal and block-respecting filters, we reconduct the filter setting to a context similar to the classical one, and we use tools similar to the ones used in the classical case, in order to prove our main results
Brooks-Jewett-type theorems for the pointwise ideal convergence of measures with values in l-groups
No full textSome Brooks-Jewett, Vitali-Hahn-Saks and Nikodym convergence-type theorems in the context of l-groups with respect to ideal convergence are proved. Moreover, an example is given, in which it is shown that in general results analogous to these kinds of limit theorems do not hold, when pointwise convergence of the measure involved is replaced by the corresponding ideal pointwise convergence
Limit theorems in l-groups with respect to D-convergence
No full textSome Schur, Vitali-Hahn-Saks and Nikodym convergence theorems for l-group-valued measures are given in the context of (D)-convergence. We consider both the sigma-additive and the finitely additive case. The pointwise convergence of the measures involved is assumed to be with respect to a common regulator, while the concepts of sigma-additivity and strong boundedness are formulated similarly as the corresponding classical ones (and not with respect to a same regulator)
Some versions of limit and Dieudonné-type theorems with respect to filter convergence for (ℓ)-group-valued measures
No full textSome limit and Dieudonné-type theorems in the setting of
l-groups with respect to filter convergence are proved, extending earlier results. A particular importance is given to (uniform) absolute continuity and (uniform) regular measures. We deal with pointwise filter convergence, and by studying "good countability properties" of filters involved we are able to reconduct some properties of ideal convergence
(which is weaker than the classical one) to some corresponding properties
of the usual convergence, in order to prove our results
Ascoli-type theorems and ideal (α)-convergence
No full textWe investigate some fundamental properties of ideal convergence and ideal exhaustiveness of real-valued function sequences, giving some characterizations of continuity of the limit function. Furthermore we establish new versions of Ascoli and Helly-type theorems, giving also some applications in measure theory
Modes of ideal continuity and the additive property in the Riesz space setting
No full textIn this paper we present some different types
of ideal convergence/divergence and of ideal continuity for Riesz space-valued functions, and prove some
basic properties and comparison results. We investigate the relations among different modes of ideal continuity and present a characterization of the -property for ideals of
an abstract set. Finally we pose some open problems
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