An 𝐿¹-space for Boolean algebras and semireflexivity of spaces 𝐿^{∞}(𝑋,Σ,𝜇)

Sentilles, Dennis

doi:10.1090/S0002-9947-1977-0430769-3

An $L^{1}$-space for Boolean algebras and semireflexivity of spaces $L^{\infty }(X,\Sigma ,\mu )$
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Trans. Amer. Math. Soc. 226 (1977), 1-37 Request permission

Abstract:

In this paper we suggest a measure free construction of ${L^1}$-spaces using Boolean algebras and strict topologies and initiate a duality theory of $({L^\infty },{L^1})$ like that of the duality of continuous functions and Baire measures, showing that the Boolean context yields a formal link between uniform tightness, uniform $\sigma$-additivity and uniform integrability.

References

Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
R. Creighton Buck, Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95–104. MR 105611
H. S. Collins, Strict, weighted, and mixed topologies and applications, Advances in Math. 19 (1976), no. 2, 207–237. MR 394149, DOI 10.1016/0001-8708(76)90062-1
Heron S. Collins, On the space $l^{\infty }\,(S)$, with the strict topology, Math. Z. 106 (1968), 361–373. MR 239406, DOI 10.1007/BF01115085
John B. Conway, The strict topology and compactness in the space of measures. II, Trans. Amer. Math. Soc. 126 (1967), 474–486. MR 206685, DOI 10.1090/S0002-9947-1967-0206685-2

La dualité entre les espaces

J. Dixmier, Sur certains espaces considérés par M. H. Stone, Summa Brasil. Math. 2 (1951), 151–182 (French). MR 48787
J. R. Dorroh, The localization of the strict topology via bounded sets, Proc. Amer. Math. Soc. 20 (1969), 413–414. MR 235412, DOI 10.1090/S0002-9939-1969-0235412-5
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge University Press, London-New York, 1974. MR 0454575, DOI 10.1017/CBO9780511897207
D. H. Fremlin, D. J. H. Garling, and R. G. Haydon, Bounded measures on topological spaces, Proc. London Math. Soc. (3) 25 (1972), 115–136. MR 344405, DOI 10.1112/plms/s3-25.1.115
Haim Gaifman, Concerning measures on Boolean algebras, Pacific J. Math. 14 (1964), 61–73. MR 161952, DOI 10.2140/pjm.1964.14.61
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
William H. Graves and Suzanne M. Molnar, A universal measure, Studies in Appl. Math. 54 (1975), no. 2, 117–134. MR 467306, DOI 10.1002/sapm1975542117
Paul R. Halmos, Lectures on Boolean algebras, Van Nostrand Mathematical Studies, No. 1, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR 0167440
J. N. Henry and D. C. Taylor, The $\bar \beta$ topology for $W^{\ast }$-algebras, Pacific J. Math. 60 (1975), no. 1, 123–139. MR 390791, DOI 10.2140/pjm.1975.60.123
J. L. Kelley, Measures on Boolean algebras, Pacific J. Math. 9 (1959), 1165–1177. MR 108570, DOI 10.2140/pjm.1959.9.1165
Dorothy Maharam, An algebraic characterization of measure algebras, Ann. of Math. (2) 48 (1947), 154–167. MR 18718, DOI 10.2307/1969222
Anthony L. Peressini, Ordered topological vector spaces, Harper & Row, Publishers, New York-London, 1967. MR 0227731
D. A. Raĭkov, A criterion of completeness of locally convex spaces, Uspehi Mat. Nauk 14 (1959), no. 1 (85), 223–229 (Russian). MR 0105000
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
Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
Stephen Scheinberg, Topologies which generate a complete measure algebra, Advances in Math. 7 (1971), 231–239 (1971). MR 286965, DOI 10.1016/S0001-8708(71)80004-X
F. Dennis Sentilles, The strict topology on bounded sets, Pacific J. Math. 34 (1970), 529–540. MR 273404, DOI 10.2140/pjm.1970.34.529
F. Dennis Sentilles, Bounded continuous functions on a completely regular space, Trans. Amer. Math. Soc. 168 (1972), 311–336. MR 295065, DOI 10.1090/S0002-9947-1972-0295065-1
F. Dennis Sentilles and Donald Curtis Taylor, Factorization in Banach algebras and the general strict topology, Trans. Amer. Math. Soc. 142 (1969), 141–152. MR 247437, DOI 10.1090/S0002-9947-1969-0247437-9
Dennis Sentilles and Robert F. Wheeler, Linear functionals and partitions of unity in $C_{b}(X)$, Duke Math. J. 41 (1974), 483–496. MR 358314
Roman Sikorski, Boolean algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 25, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0126393
V. S. Varadarajan, Measures on topological spaces, Mat. Sb. (N.S.) 55 (97) (1961), 35–100 (Russian). MR 0148838