Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Extensions of the $ v$-integral

Authors: J. R. Edwards and S. G. Wayment
Journal: Trans. Amer. Math. Soc. 191 (1974), 165-184
MSC: Primary 28A25; Secondary 28A45
MathSciNet review: 0349941
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Abstract: In Representations for transformations continuous in the BV norm [J. R. Edwards and S. G. Wayment, Trans. Amer. Math. Soc. 154 (1971), 251-265] the $ \nu$-integral is defined over intervals in $ {E^1}$ and is used to give a representation for transformations continuous in the BV norm. The functions f considered therein are real valued or have values in a linear normed space X, and the transformation $ T(f)$ is real or has values in a linear normed space Y. In this paper the $ \nu$-integral is extended in several directions: (1) The domain space to (a) $ {E^n}$, (b) an arbitrary space S, a field $ \Sigma $ of subsets of S and a bounded positive finitely additive set function $ \mu $ on $ \Sigma $ (in this setting the function space is replaced by the space of finitely additive set functions which are absolutely continuous with respect to $ \mu $); (2) the function space to (a) bounded continuous, (b) $ {C_c}$, (c) $ {C_0}$, (d) C with uniform convergence on compact sets; (3) range space X for the functions and Y for the transformation to topological vector spaces (not necessarily convex); (4) when X and Y are locally convex spaces, then a representation for transformations on a $ {C_1}$-type space of continuously differentiable functions with values in X is given.

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Keywords: Generalized Stieltjes integral, $ \nu$-integral, convex space, topological vector space, bounded variation, bounded semivariation, weak bounded variation, spaces of absolutely continuous functions, spaces of continuous functions, Lebesgue-type spaces of functions, spaces of continuously differentiable functions, convex with respect to area, $ \tau $-quasi-Gowurin, fundamental function, integral representation
Article copyright: © Copyright 1974 American Mathematical Society