Weak integral convergence theorems and operator measures (original) (raw)

Topological Dual Systems for Spaces of Vector Measurep-Integrable Functions

Journal of Function Spaces

We show a Dvoretzky-Rogers type theorem for the adapted version of theq-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.

Convergence for varying measures in the topological case

Annali di Matematica Pura ed Applicata (1923 -)

In this paper convergence theorems for sequences of scalar, vector and multivalued Pettis integrable functions on a topological measure space are proved for varying measures vaguely convergent.

A Note on Radon-Nikodym Theorem for Operator Valued Measures and Its Applications

Communications of the Korean Mathematical Society

In this note we present sufficient conditions for the existence of Radon-Nikodym derivatives (RND) of operator valued measures with respect to scalar measures. The RND is characterized by the Bochner integral in the strong operator topology of a strongly measurable operator valued function with respect to a nonnegative finite measure. Using this result we also obtain a characterization of compact sets in the space of operator valued measures. An extension of this result is also given using the theory of Pettis integral. These results have interesting applications in the study of evolution equations on Banach spaces driven by operator valued measures as structural controls.

A note on convergence in measure and selection principles

Filomat, 2012

It is proved that some classes of sequences of measurable functions satisfy certain selection principles related to special modes of convergence (convergence in measure, almost everywhere convergence, almost uniform convergence, mean convergence).

Weak convergences of probability measures: A uniform principle

1998

We consider a set of probability measures on a locally compact separable metric space. It is shown that a necessary and sufficient condition for (relative) sequential compactness of in various weak topologies (among which the vague, weak and setwise topologies) has the same simple form; i.e. a uniform principle has to hold in. We also extend this uniform principle to some Köthe function spaces.

Weak convergence in spaces of measures and operators

Bulletin of the Belgian Mathematical Society - Simon Stevin, 1999

J. K. Brooks and P. W. Lewis have established that if E and E * have RNP, then in M (Σ, E), m n converges weakly to m if and only if m n (A) converges weakly to m(A) for each A ∈ Σ. Assuming the existence of a special kind of lifting, N. Randrianantoanina and E. Saab have shown an analogous result if E is a dual space. Here we show that for the space M (P(N), E) where E * is a Grothendieck space or E is a Mazur space, this kind of weak convergence is valid. Also some applications for subspaces of L(E, F) similar to the results of N. Kalton and W. Ruess are given.

A note on the weak convergence of continuously integrable sequences

2021

Abstract. A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the σ(L, L) topology. In this paper, we link such a result to weak convergence theory of bounded measures as exposed in Billingsley (1968) and in Lo(2021) to offer a detailed and new proof using the ideas beneath the proof of prohorov’s theorem where the continuous integrability replaces the uniform or asymptotic tightness.

Strongly Bounded Representing Measures and Convergence Theorems

Glasgow Mathematical Journal, 2010

Let K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).