On Pettis integrals with separable range. (English) Zbl 0823.28005

Plebanek, Grzegorz

Colloq. Math. 64, No. 1, 71-78 (1993).

Using techniques developed in G. A. Edgar [Indiana Univ. Math. J. 26, 663-677 (1977; Zbl 0361.46017); ibid. 28, 559-579 (1979; Zbl 0418.46034)] and M. Talagrand [Mem. Am. Math. Soc. 307 (1984; Zbl 0582.46049)], the author studies Pettis integrability and Pettis integrals with separable range. For a weak Baire measure \(\lambda\) on a Banach space \(E\), \(\lambda\) is said to be scalarly concentrated on a subspace \(G\) of \(E\) if \(x^* |_ G= 0\) implies \(x^*= 0\), \(\lambda\)-almost everywhere. Then the author shows that for an \(E\)-valued weakly measurable and scalarly bounded function \(f\) defined on a finite measure space \((\Omega, \Sigma, \mu)\), \(f\) is Pettis integrable and its Pettis integral has separable range if and only if the weak Baire measure \(f(\mu)\) (the image measure of \(\mu\) by \(f\)) is scalarly concentrated on a separable subspace of \(E\). Its proof is based on a lemma which is a version of Lemma 5-1-2 in M. Talagrand’s work quoted above, and this lemma also yields a sequential completeness of the spaces of Grothendieck measures on completely regular Hausdorff spaces. Moreover, he presents two results on Pettis integrability and the range of Pettis integrals in \(C(K)\) (the Banach space of continuous functions defined on a compact Hausdorff space \(K\)).

Reviewer: M.Matsuda (Ohya/Shizuoka)

MSC:

28B05	Vector-valued set functions, measures and integrals
46G10	Vector-valued measures and integration

Keywords:

Pettis integrals; separable range; scalarly bounded function; weak Baire measure; Grothendieck measures; completely regular Hausdorff spaces

Citations:

Zbl 0361.46017; Zbl 0418.46034; Zbl 0582.46049