Space Of Pettis Integrable Functions Research Articles

Correction to the paper ‘Copies of $\ell _{\infty }$ in the space of Pettis integrable functions with integrals of finite variation’ (Studia Math. 210 (2012), 93–98)

Correction to the paper ‘Copies of $\ell _{\infty }$ in the space of Pettis integrable functions with integrals of finite variation’ (Studia Math. 210 (2012), 93–98)

Copies of c 0 in the space of Pettis integrable functions with integrals of finite variation

Let (Ω,Σ,μ) be a complete finite measure space and X a Banach space. If all X-valued Pettis integrals defined on (Ω,Σ,μ) have separable ranges we show that the space of all weakly μ-measurable (classes of scalarly equivalent) X-valued Pettis integrable functions with integrals of finite variation, equipped with the variation norm, contains a copy of c0 if and only if X does.

Integral Representations for a Class of Operators on <em>L</em><sup>1</sup><sub><em>E</em></sub>

Let \((X, \mathcal{A}, \mu)\) be a finite measure space, \(E\) a locally convex Hausdorff space, \(L_{E}^{1}\) the space of functions \(f: X \to E\) which are \(\mu\)-integrable by semi-norms, \(P(\mu, E)\) the space of Pettis integrable functions and \(P_{1}(\mu, E)\) those elements of \(P(\mu, E)\) which are measurable by semi-norms. We prove that a linear continuous mapping \( T: L_{E}^{1} \to E\) is of the form \(T(f)= \int g f d \mu\) (\(g \in L^{\infty}\)) if and only if \( h( T(f))=0\) whenever \( h\circ f=0 \) for any \( f \in L_{E}^{1}, h \in E'\). Similar results are proved for \(P(\mu, E)\) and \(P_{1}(\mu, E)\).

Decomposability and uniform integrability in Pettis integration

In this paper, we give several characterizations of decomposable subsets of the space of Pettis integrable functions (in particular, a characterization similar to that already known in the Bochner integrability setting). We also introduce the notion WPUI of uniform integrability, and obtain some relations between decomposability and PUI, resp. WPUI uniform integrability concepts. As consequence, conditional weak compactness and sequential weak compactness criteria are given under decomposability assumptions. Finally, an application to second order differential inclusions is presented.

Correction to the paper "Embedding c 0 in the space of Pettis integrable functions"

A complete argument is given for the proof of Theorem 5 in [F].

A GLIDING HUMP PROPERTY AND BANACH-MACKEY SPACES

We consider the Banach–Mackey property for pairs of vector spaces E and E0 which are in duality. Let A be an algebra of sets and assume that P is an additive map from A into the projection operators on E. We define a continuous gliding hump property for the map P and show that pairs with this gliding hump property and another measure theoretic property are Banach-Mackey pairs, i. e., weakly bounded subsets of E are strongly bounded. Examples of vector valued function spaces, such as the space of Pettis integrable functions, which satisfy these conditions are given.

A MULTIPLIER GLIDING HUMP PROPERTY FOR SEQUENCE SPACES

We consider the Banach-Mackey property for pairs of vector spaces E and E0 which are in duality. Let A be an algebra of sets and assume that P is an additive map from A into the projection operators on E. We define a continuous gliding hump property for the map P and show that pairs with this gliding hump property and another measure theoretic property are Banach-Mackey pairs,i.e., weakly bounded subsets of E are strongly bounded. Examples of vector valued function spaces, such as the space of Pettis integrable functions, which satisfy these conditions are given.

Sets which are dense in the space of all Pettis integrable functions

Using the examples given by S.J. Dilworth and M. Girardi, we prove that the set of all nowhere Pettis differentiable functions is a G δ -dense set in the space of all X-valued Pettis integrable functions on [0, 1]. Also an another example of a dense set in the space of all Pettis integrable functions is given.

Precompactness in the space of Pettis integrable functions

In this paper we give a necessary and sufficient condition for weak precompactness in the space of Pettis integrable functions with relatively compact range of their indefinite integrals. The main result is formulated in the spirit of previous works by J. K. Brooks and N. Dinculeanu [ 1,2], where such a theorem was obtained for strongly measurable and Pettis integrable functions. Also we show that, in a sense, our result is the best possible (see Remark 2).

COMPLEMENTED COPIES OFc0IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract Let X be a Banach space containing a copy of c0, then the space of Pettis integrable functions defined from any perfect atomless measure space to X, contains a complemented copy of c0.

The Space of Pettis Integrable Functions is Barrelled

It is well known that the normed space of Pettis integrable functions from a finite measure space to a Banach space is not complete in general. Here we prove that this space is always barrelled; this tells us that we may apply two important results to this space, namely, the Banach-Steinhaus uniform boundedness principle and the closed graph theorem. The proof is based on a theorem stating that a quasi-barrelled space having a convenient Boolean algebra of projections is barrelled. We also use this theorem to give similar results for the spaces of Bochner integrable functions.

The space of Pettis integrable functions is barrelled

It is well known that the normed space of Pettis integrable functions from a finite measure space to a Banach space is not complete in general. Here we prove that this space is always barrelled; this tells us that we may apply two important results to this space, namely, the Banach-Steinhaus uniform boundedness principle and the closed graph theorem. The proof is based on a theorem stating that a quasi-barrelled space having a convenient Boolean algebra of projections is barrelled. We also use this theorem to give similar results for the spaces of Bochner integrable functions.

On weak compactness in the space of pettis integrable functions

On weak compactness in the space of pettis integrable functions