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

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

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.

**[1]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****[2]**R. J. Easton and D. H. Tucker,*A generalized lebesgue-type integral*, Math. Ann.**181**(1969), no. 4, 311–324. MR**1513279**, https://doi.org/10.1007/BF01350670**[3]**R. J. Easton and S. G. Wayment,*A mean value theorem*, Amer. Math. Monthly**77**(1970), 170–172. MR**0255759**, https://doi.org/10.2307/2317334**[4]**R. J. Easton, D. H. Tucker, and S. G. Wayment,*On the existence almost everywhere of the cross partial derivatives*, Math. Z.**102**(1967), 171–176. MR**0218502**, https://doi.org/10.1007/BF01112436**[5]**J. R. Edwards and S. G. Wayment,*A 𝑣-integral representation for linear operators on spaces of continuous functions with values in topological vector spaces*, Pacific J. Math.**35**(1970), 327–330. MR**0274703****[6]**J. R. Edwards and S. G. Wayment,*A 𝑣-integral representation for the continuous linear operators on spaces of continuously differentiable vector-valued functions*, Proc. Amer. Math. Soc.**30**(1971), 263–270. MR**0281031**, https://doi.org/10.1090/S0002-9939-1971-0281031-3**[7]**J. R. Edwards and S. G. Wayment,*A unifying representation theorem*, Math. Ann.**187**(1970), 317–328. MR**0270181**, https://doi.org/10.1007/BF01396462**[8]**J. R. Edwards and S. G. Wayment,*Integral representations for continuous linear operators in the setting of convex topological vector spaces*, Trans. Amer. Math. Soc.**157**(1971), 329–345. MR**0281867**, https://doi.org/10.1090/S0002-9947-1971-0281867-3**[9]**J. R. Edwards and S. G. Wayment,*Representations for transformations continuous in the 𝐵𝑉 norm*, Trans. Amer. Math. Soc.**154**(1971), 251–265. MR**0274704**, https://doi.org/10.1090/S0002-9947-1971-0274704-4**[10]**Charles Fefferman,*A Radon-Nikodym theorem for finitely additive set functions*, Pacific J. Math.**23**(1967), 35–45. MR**0215956****[11]**Robert K. Goodrich,*A Riesz representation theorem in the setting of locally convex spaces*, Trans. Amer. Math. Soc.**131**(1968), 246–258. MR**0222681**, https://doi.org/10.1090/S0002-9947-1968-0222681-4**[12]**D. H. Tucker,*An existence theorem for a Goursat problem*, Pacific J. Math.**12**(1962), 719–727. MR**0144075****[13]**Don H. Tucker,*A representation theorem for a continuous linear transformation on a space of continuous functions*, Proc. Amer. Math. Soc.**16**(1965), 946–953. MR**0199722**, https://doi.org/10.1090/S0002-9939-1965-0199722-9**[14]**David J. Uherka,*Generalized Stieltjes integrals and a strong representation theorem for continuous linear maps on a function space*, Math. Ann.**182**(1969), 60–66. MR**0247439**, https://doi.org/10.1007/BF01350164

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Additional Information

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