Extensions of the integral
Authors:
J. R. Edwards and S. G. Wayment
Journal:
Trans. Amer. Math. Soc. 191 (1974), 165184
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), 251265] the integral is defined over intervals in 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 is real or has values in a linear normed space Y. In this paper the integral is extended in several directions: (1) The domain space to (a) , (b) an arbitrary space S, a field of subsets of S and a bounded positive finitely additive set function on (in this setting the function space is replaced by the space of finitely additive set functions which are absolutely continuous with respect to ); (2) the function space to (a) bounded continuous, (b) , (c) , (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 type space of continuously differentiable functions with values in X is given.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403499413
PII:
S 00029947(1974)03499413
Keywords:
Generalized Stieltjes integral,
integral,
convex space,
topological vector space,
bounded variation,
bounded semivariation,
weak bounded variation,
spaces of absolutely continuous functions,
spaces of continuous functions,
Lebesguetype spaces of functions,
spaces of continuously differentiable functions,
convex with respect to area,
quasiGowurin,
fundamental function,
integral representation
Article copyright:
© Copyright 1974
American Mathematical Society
