Pettis integration via the Stonian transform

Sentilles, F. Dennis; Wheeler, Robert F.

doi:10.2140/pjm.1983.107.473

Cited by 14 publications

(6 citation statements)

References 22 publications

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“…It follows from the Hahn-Banach Theorem that in this case {co <p(S\B): B e 2, n{B) = 0} (see [14, 2.2] and [23]). The intersection of this set with E is known as the core of <1> relative to S (see [1, p. 311]).…”

Section: Let E and F Be Locally Convex Spaces With Ementioning

confidence: 99%

Extending Edgar's ordering to locally convex spaces

Robertson

1992

Glasgow Math. J.

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Section: Let E and F Be Locally Convex Spaces With Ementioning

confidence: 99%

Extending Edgar's ordering to locally convex spaces

Robertson

1992

Glasgow Math. J.

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“…The following reformulation of Proposition 1 was derived from ideas in proofs due to M. Talagrand (see Sentilles and Wheeler [5]). PROPOSITION 3.…”

Section: Proposition 1 a Dunford Integrable Function F Is Pettis Intmentioning

confidence: 99%

Remarks on Pettis integrability

Huff¹

1986

Proc. Amer. Math. Soc.

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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 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].

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“…The Pettis integrability of a bounded weakly measurable function f.S-* X is analysed in [9] via the Stonian transform/: K -> X**, defined on the Stone representation space K of the measure algebra B a {S)/n~\0). Given/in C%(S, X), it seems more appropriate to consider the canonical extension^ :/?S -• X**, defined in a similar way: <x*,/ /?…”

Section: Integrability Of a Bounded Weakly Continuous Function Via Itmentioning

confidence: 99%

Pettis Integrability of Weakly Continuous Functions and Baire Measures

Pallarés

Vera

1985

Journal of the London Mathematical Society

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We analyse the Pettis integrability of weakly continuous bounded functions defined on a completely regular space S and taking values in a Banach space. We prove that the set of Baire measures with respect to which such functions are universally Pettis integrable is precisely the space M g (S) of Grothendieck measures introduced by Wheeler. This leads us to prove that M g (S) is a(M g (S), C 6 (5))-sequentially complete, and we obtain a characterization in 0S of the measures in M g (S). We also obtain analogous results for the space of separable measures M w (5).

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Pettis integration via the Stonian transform

Cited by 14 publications

References 22 publications

Extending Edgar's ordering to locally convex spaces

Extending Edgar's ordering to locally convex spaces

Remarks on Pettis integrability

Pettis Integrability of Weakly Continuous Functions and Baire Measures

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