Extensions of the -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 -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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DOI:
https://doi.org/10.1090/S0002-9947-1974-0349941-3

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,
Lebesgue-type spaces of functions,
spaces of continuously differentiable functions,
convex with respect to area,
-quasi-Gowurin,
fundamental function,
integral representation

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© Copyright 1974
American Mathematical Society