An isomorphism and isometry theorem for a class of linear functionals
Author:
William D. L. Appling
Journal:
Trans. Amer. Math. Soc. 199 (1974), 131140
MSC:
Primary 28A25
MathSciNet review:
0352385
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Abstract: Suppose is a set, is a field of subsets of and is the set of all realvalued, 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 abovementioned isomorphism and isometry theorem, together with a previous representation theorem of the author (J. London Math. Soc. 44 (1969), pp. 385396), imply an analogue of a dual representation theorem of Edwards and Wayment (Trans. Amer. Math. Soc. 154 (1971), pp. 251265). Finally, a ``pseudorepresentation 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 (34 #4446), http://dx.doi.org/10.1090/S00029939196702046077
 [2]
William
D. L. Appling, Summability of realvalued set functions, Riv.
Mat. Univ. Parma (2) 8 (1967), 77–100. MR 0251187
(40 #4418)
 [3]
William
D. L. Appling, Concerning a class of linear transformations,
J. London Math. Soc. 44 (1969), 385–396. MR 0237734
(38 #6015)
 [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
(43 #466), http://dx.doi.org/10.1090/S00029947197102747044
 [6]
A. Kolmogoroff, Untersuchen über den Integralbegriff, Math. Ann. 103 (1930), 654696.
 [7]
D. Mauldin, An integral representation of functionals on , preliminary report, Notices Amer. Math. Soc. 18 (1971), 949. Abstract #71TB198.
 [1]
 W. D. L. Appling, Some integral characterizations of absolute continuity, Proc. Amer. Math. Soc. 18 (1967), 9499. MR 34 #4446. MR 0204607 (34:4446)
 [2]
 , Summability of realvalued set functions, Riv. Mat. Univ. Parma (2) 8 (1967), 77100. MR 40 #4418. MR 0251187 (40:4418)
 [3]
 , Concerning a class of linear transformations, J. London Math. Soc. 44 (1969), 385396. MR 38 #6015. MR 0237734 (38:6015)
 [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), 251265. MR 43 #466. MR 0274704 (43:466)
 [6]
 A. Kolmogoroff, Untersuchen über den Integralbegriff, Math. Ann. 103 (1930), 654696.
 [7]
 D. Mauldin, An integral representation of functionals on , preliminary report, Notices Amer. Math. Soc. 18 (1971), 949. Abstract #71TB198.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403523851
PII:
S 00029947(1974)03523851
Keywords:
integral,
isomorphism,
isometry,
transformation representation
Article copyright:
© Copyright 1974 American Mathematical Society
