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On the product of vector measures

Published online by Cambridge University Press:  09 April 2009

Igor Kluvánek
Affiliation:
University of Illinois, Urbana, Illinois, U.S.A.
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Let μ and ν be measures defined on some σ-algebrs with values in locally convex topological vector spaces X and Y, repectively. It is possible [1] to construct their product λ = μ × ν as a measure on a σ-algebra if λ is allowed to take its values in X ⊗ε Y, the completion of X ⊗ Y in the topology of bi-equicontinuous convergence. The reason is, roughly speaking, that the topology of biequicontinuous convergence on X ⊗ Y is coarse enough to make λ σ-additive and the completion X ⊗ε Y is big enough to accommodate all values of λ. Here we are going to improve the result by introducing a finer topology on X ⊗ Y in which λ will be σ-additive and such that all values of λ will belong to the completion of X ⊗ Y under that topology. The topology in question is obtained by a slight modification from a topology considered for the first time in the work [3] of Jacobs. Curiously enough, the proof of the improved result is simpler than that of [1] and reduces almost to a direct observation avoiding duality arguments.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Duchoň, M. and Kluvánek, I.. ‘Inductive tensor product of vector-valued measures’, Matematický Časopis 17 (1967), 108111.Google Scholar
[2]Dunford, N. and Schwartz, J. T., Linear Operators I (New York, 1958).Google Scholar
[3]Jacobs, H., Ordered topological tensor products (Thesis, University of Illinois, Urbana 1969).Google Scholar
[4]Kluvánek, I., ‘Contribution to the theory of vector measures’ (Russian), Matematiko-fyzikálny časopis 11 (1961), 173191.Google Scholar
[5]Kluvánek, I., ‘Contribution to the theory of vector measures II’ (Russian), Matematicko-fyzikálny časopis 16 (1966), 7681.Google Scholar