Complete topologies on spaces of Baire measure
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- by R. B. Kirk PDF
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Abstract:
Let X be a completely regular Hausdorff space, let L be the linear space of all finite linear combinations of the point measures on X and let ${M_\sigma }$ denote the space of Baire measures on X. The following is proved: If ${M_\sigma }$ is endowed with the topology of uniform convergence on the uniformly bounded, equicontinuous subsets of ${C^b}(X)$, then ${M_\sigma }$ is a complete locally convex space in which L is dense and whose dual is ${C^b}(X)$, provided there are no measurable cardinals. A complete description of the situation in the presence of measurable cardinals is also given. Let ${M_C}$ be the subspace of ${M_\sigma }$ consisting of those measures which have compact support in the realcompactification of X. The following result is proved: If ${M_C}$ is endowed with the topology of uniform convergence on the pointwise bounded and equicontinuous subsets of $C(X)$, then ${M_C}$ is a complete locally convex space in which L is dense and whose dual is $C(X)$, provided there are no measurable cardinals. Again the situation if measurable cardinals exist is described completely. Let M denote the Banach dual of ${C^b}(X)$. The following is proved: If M is endowed with the topology of uniform convergence on the norm compact subsets of ${C^b}(X)$, then M is a complete locally convex space in chich L is dense. It is also proved that ${M_\sigma }$ is metrizable if and only if X is discrete and that the metrizability of either ${M_C}$ or M is equivalent to X being finite. Finally the following is proved: If ${M_C}$ has the Mackey topology for the pair $({M_C},C(X))$, then ${M_C}$ is complete and L is dense in ${M_C}$.References
- A. D. Alexandroff, Additive set-functions in abstract spaces, Rec. Math. [Mat. Sbornik] N. S. 8 (50) (1940), 307–348 (English, with Russian summary). MR 0004078
- Richard F. Arens and James Eells Jr., On embedding uniform and topological spaces, Pacific J. Math. 6 (1956), 397–403. MR 81458, DOI 10.2140/pjm.1956.6.397
- R. M. Dudley, Convergence of Baire measures, Studia Math. 27 (1966), 251–268. MR 200710, DOI 10.4064/sm-27-3-251-268
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
- G. G. Gould and M. Mahowald, Measures on completely regular spaces, J. London Math. Soc. 37 (1962), 103–111. MR 156191, DOI 10.1112/jlms/s1-37.1.103
- E. Granirer, On Baire measures on $D$-topological spaces, Fund. Math. 60 (1967), 1–22. MR 208355, DOI 10.4064/fm-60-1-1-22
- Edwin Hewitt, Linear functionals on spaces of continuous functions, Fund. Math. 37 (1950), 161–189. MR 42684, DOI 10.4064/fm-37-1-161-189
- M. Katětov, Measures in fully normal spaces, Fund. Math. 38 (1951), 73–84. MR 48531, DOI 10.4064/fm-38-1-73-84
- M. Katětov, On a category of spaces, General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961) Academic Press, New York; Publ. House Czech. Acad. Sci., Prague, 1962, pp. 226–229. MR 0187190 —, On certain projectively generated continuity structures, Celebrazioni archimedee de secolo, Simposio di topologia, 1964, pp. 47-50.
- M. Katětov, Projectively generated continuity structures: A correction, Comment. Math. Univ. Carolinae 6 (1965), 251–255. MR 219027
- R. B. Kirk, Measures in topological spaces and $B$-compactness, Nederl. Akad. Wetensch. Proc. Ser. A 72=Indag. Math. 31 (1969), 172–183. MR 0246104, DOI 10.1016/1385-7258(69)90007-9
- R. B. Kirk, Locally compact, $B$-compact spaces, Nederl. Akad. Wetensch. Proc. Ser. A 72=Indag. Math. 31 (1969), 333–344. MR 0264609, DOI 10.1016/1385-7258(69)90034-1
- R. B. Kirk, Algebras of bounded real-valued functions. I, Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math. 34 (1972), 443–451. MR 0320725, DOI 10.1016/1385-7258(72)90041-8 —, A note on the Mackey topology for $({C^b}{(X)^\ast },{C^b}(X))$, Pacific J. Math. (to appear).
- J. D. Knowles, Measures on topological spaces, Proc. London Math. Soc. (3) 17 (1967), 139–156. MR 204602, DOI 10.1112/plms/s3-17.1.139 G. Köthe, Topologische linear Räume. I, Die Grundlehren der math. Wissenschaften, Band 107, Springer-Verlag, Berlin, 1960; English transl., Die Grundlehren der math. Wissenschaften, Band 159, Springer-Verlag, New York, 1969. MR 24 #A411; 40 #1750.
- W. Moran, The additivity of measures on completely regular spaces, J. London Math. Soc. 43 (1968), 633–639. MR 228645, DOI 10.1112/jlms/s1-43.1.633
- W. Moran, Measures on metacompact spaces, Proc. London Math. Soc. (3) 20 (1970), 507–524. MR 437706, DOI 10.1112/plms/s3-20.3.507
- E. Marczewski and R. Sikorski, Measures in non-separable metric spaces, Colloq. Math. 1 (1948), 133–139. MR 25548, DOI 10.4064/cm-1-2-133-139
- John S. Pym, Positive functionals, additivity, and supports, J. London Math. Soc. 39 (1964), 391–399. MR 165357, DOI 10.1112/jlms/s1-39.1.391
- Vlastimil Pták, Algebraic extensions of topological spaces, Contributions to Extension Theory of Topological Structures (Proc. Sympos., Berlin, 1967) Deutscher Verlag Wissensch., Berlin, 1969, pp. 179–188. MR 0247415
- D. A. Raĭkov, Free locally convex spaces for uniform spaces, Mat. Sb. (N.S.) 63(105) (1964), 582–590 (Russian). MR 0162120 F. D. Sentilles, and R. F. Wheeler, Additivity of functionals and the strict topology (unpublished).
- V. S. Varadarajan, Measures on topological spaces, Mat. Sb. (N.S.) 55 (97) (1961), 35–100 (Russian). MR 0148838
- A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. MR 0162118
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 184 (1973), 1-29
- MSC: Primary 28A32; Secondary 60B05
- DOI: https://doi.org/10.1090/S0002-9947-1973-0325913-9
- MathSciNet review: 0325913