An isomorphism and isometry theorem for a class of linear functionals

Author:
William D. L. Appling

Journal:
Trans. Amer. Math. Soc. **199** (1974), 131-140

MSC:
Primary 28A25

DOI:
https://doi.org/10.1090/S0002-9947-1974-0352385-1

MathSciNet review:
0352385

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Abstract: Suppose is a set, is a field of subsets of and is the set of all real-valued, finitely additive functions defined on . Two principal notions are considered in this paper. The first of these is that of a subset of , defined by certain closure properties and called a -set. The second is that of a collection of linear transformations from into with special boundedness properties. Given a -set which is a linear space, an isometric isomorphism is established from the dual of onto the set of all elements of with range a subset of . As a corollary it is demonstrated that the above-mentioned isomorphism and isometry theorem, together with a previous representation theorem of the author (J. London Math. Soc. **44** (1969), pp. 385-396), imply an analogue of a dual representation theorem of Edwards and Wayment (Trans. Amer. Math. Soc. **154** (1971), pp. 251-265). Finally, a ``pseudo-representation theorem'' for the dual of is demonstrated.

**[1]**William D. L. Appling,*Some integral characterizations of absolute continuity*, Proc. Amer. Math. Soc.**18**(1967), 94–99. MR**0204607**, https://doi.org/10.1090/S0002-9939-1967-0204607-7**[2]**William D. L. Appling,*Summability of real-valued set functions*, Riv. Mat. Univ. Parma (2)**8**(1967), 77–100. MR**0251187****[3]**William D. L. Appling,*Concerning a class of linear transformations*, J. London Math. Soc.**44**(1969), 385–396. MR**0237734**, https://doi.org/10.1112/jlms/s1-44.1.385**[4]**-,*A generalization of absolute continuity and of an analogue of the Lebesgue decomposition theorem*, Riv. Mat. Univ. Parma (to appear).**[5]**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**[6]**A. Kolmogoroff,*Untersuchen über den Integralbegriff*, Math. Ann.**103**(1930), 654-696.**[7]**D. Mauldin,*An integral representation of functionals on*, preliminary report, Notices Amer. Math. Soc.**18**(1971), 949. Abstract #71T-B198.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0352385-1

Keywords:
-integral,
isomorphism,
isometry,
transformation representation

Article copyright:
© Copyright 1974
American Mathematical Society