An isomorphism and isometry theorem for a class of linear functionals
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- by William D. L. Appling PDF
- Trans. Amer. Math. Soc. 199 (1974), 131-140 Request permission
Abstract:
Suppose $U$ is a set, ${\mathbf {F}}$ is a field of subsets of $U$ and ${\mathfrak {p}_{AB}}$ is the set of all real-valued, finitely additive functions defined on ${\mathbf {F}}$. Two principal notions are considered in this paper. The first of these is that of a subset of ${\mathfrak {p}_{AB}}$, defined by certain closure properties and called a $C$-set. The second is that of a collection $\mathcal {C}$ of linear transformations from ${\mathfrak {p}_{AB}}$ into ${\mathfrak {p}_{AB}}$ with special boundedness properties. Given a $C$-set $M$ which is a linear space, an isometric isomorphism is established from the dual of $M$ onto the set of all elements of $\mathcal {C}$ with range a subset of $M$. 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 ${\mathfrak {p}_{AB}}$ is demonstrated.References
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- William D. L. Appling, Summability of real-valued set functions, Riv. Mat. Univ. Parma (2) 8 (1967), 77–100. MR 251187
- William D. L. Appling, Concerning a class of linear transformations, J. London Math. Soc. 44 (1969), 385–396. MR 237734, DOI 10.1112/jlms/s1-44.1.385 —, A generalization of absolute continuity and of an analogue of the Lebesgue decomposition theorem, Riv. Mat. Univ. Parma (to appear).
- J. R. Edwards and S. G. Wayment, Representations for transformations continuous in the $\textrm {BV}$ norm, Trans. Amer. Math. Soc. 154 (1971), 251–265. MR 274704, DOI 10.1090/S0002-9947-1971-0274704-4 A. Kolmogoroff, Untersuchen über den Integralbegriff, Math. Ann. 103 (1930), 654-696. D. Mauldin, An integral representation of functionals on $ca(S,\Sigma )$, preliminary report, Notices Amer. Math. Soc. 18 (1971), 949. Abstract #71T-B198.
Additional Information
- © Copyright 1974 American Mathematical Society
- 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