Remarks on Pettis integrability

Abstract

Characterizations of Pettis integrability, including the Geitz-Talagrand core theorem, are derived in an easy way. The purpose of this note is to point out how a folklore result (Proposition 1) can be made the basis for relatively easy proofs of some recent results about Pettis integrability. Our notation follows Dunford and Schwartz [1]. Let (fi, E, A) be a complete probability space, and let X be a Banach space with continuous dual X*. A function /: Í2-► X is Dunford integrable provided the composition T{x*) = x*f is in L1(X) for every x* in X*. In this case, it follows (from the closed graph theorem) that T:X*-► Ll(X) is a bounded linear operator. Hence, for every g in 7/°°(A), the map pg, defined by <pg(x*) = I gT(x*)dX, is in X**. In particular, for each E in £, u{E) = JE f dX, defined to equal <çXE and called the Dunford integral of / over E, is an element of X**. The function v: £-> X** is not necessarily countably additive. It can be shown that v is countably additive if and only if T is a weakly compact operator if and only if {x*f: \\x*\\ < 1} is uniformly integrable in L1{X) [1, pp. 319, 485, 292].

Higher moments of Banach space valued random variables

“…However, we will not find much use of this, since we will take integrals in tensor products By the definition (and discussion) above, X is Dunford integrable if and only if x˚Þ Ñ x˚pXq is a bounded linear operator B˚Ñ L 1 pPq. If X is Pettis integrable, then furthermore this operator is weakly compact, i.e., it maps the unit ball into a relatively weakly compact subset of L 1 pPq, see [62], [36] or [57,Proposition 3.7] [36]). If tx˚pXq : x˚P B˚, }x˚} ď 1u is uniformly integrable and X is weakly a.s. separably valued, then X is Pettis integrable.…”

“…Remark 5.9. Actually, Huff [36] uses a condition that he calls separablelike; the definition given in [36] is somewhat stronger than weak a.s. separability, but it seems likely that he really intended what we call weak a.s. separability, and the proof in [36] uses only weak a.s. separability. See also Stefánsson [59] where weakly a.s. separably valued is called determined by a separable subspace and said to be the same as Huff's separable-like.…”

“…By the definition (and discussion) above, X is Dunford integrable if and only if x ˚Þ Ñ x ˚pX q is a bounded linear operator B ˚Ñ L 1 pPq. If X is Pettis integrable, then furthermore this operator is weakly compact, i.e., it maps the unit ball into a relatively weakly compact subset of L 1 pPq, see [62], [36] or [57,Proposition 3.7]. A subset of L [36] has shown that it holds if X is weakly a.s. separably valued.…”

“…If X is Pettis integrable, then furthermore this operator is weakly compact, i.e., it maps the unit ball into a relatively weakly compact subset of L 1 pPq, see [62], [36] or [57,Proposition 3.7]. A subset of L [36] has shown that it holds if X is weakly a.s. separably valued. (In particular, the converse holds when B is separable.)…”

Young Measures on Topological Spaces

“…Some parts of the results presented here are borrowed from [ACV92, ACV98, BCG99b, BC01], see also [BS03]. If f : Ω → E is a scalarly integrable function, the Pettis norm f Pe of f [Gei81, Huf86,Mus91] is defined by f Pe = sup x ∈B E * Ω | x , f | d P. The space P 1 E (Ω, S, P) of E-valued Pettis integrable functions is endowed with the Pettis norm . Pe .…”