Extensions of the 𝑣-integral

Edwards, J. R.; Wayment, S. G.

doi:10.1090/S0002-9947-1974-0349941-3

Extensions of the $v$-integral
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by J. R. Edwards and S. G. Wayment PDF

Trans. Amer. Math. Soc. 191 (1974), 165-184 Request permission

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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