The product of vector-valued measures

Publisher: Cambridge University Press

E-ISSN: 1755-1633|8|3|359-366

ISSN: 0004-9727

Source: Bulletin of the Australian Mathematical Society, Vol.8, Iss.3, 1973-06, pp. : 359-366

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

Let M (N) be a σ–algebra of subsets of a set S (T) and let X, Y be Banach spaces with (,) a continuous bilinear map from X × Y into the scalar field. If μ: MX (v: NY) is a vector measure and λ is the scalar measure defined on the measurable rectangles A × B, A M, B N, by λ(A×B) = μ(A), v(B), it is known that λ is generally not countably additive on the algebra generated by the measurable rectangles and therefore has no countably additive extension to the σ-algebra generated by the measurable rectangles. If μ (v) is an indefinite Pettis integral it is shown that a necessary and sufficient condition that λ have a countable additive extension to the σ-algebra generated by the measurable rectangles is that the function F: (s, t) → f(s), g(t) is integrable with respect to α × β.