Lebesgue’s theorem of differentiation in Fréchet lattices

Grosse-Erdmann, Karl-Goswin

doi:10.1090/S0002-9939-1991-1062390-3

Lebesgue’s theorem of differentiation in Fréchet lattices
HTML articles powered by AMS MathViewer

by Karl-Goswin Grosse-Erdmann PDF

Proc. Amer. Math. Soc. 112 (1991), 371-379 Request permission

Abstract:

Lebesgue’s differentiation theorem (LDT) states that every monotonic real function is differentiable a.e. We investigate the validity of this theorem for functions with values in topological vector lattices. It is shown that a Fréchet lattice satisfies (LDT) iff it is isomorphic to a generalized echelon space, a Banach lattice satisfies (LDT) iff it is isomorphic to some ${l^1}\left ( \Gamma \right )$.

References

J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
L. Egghe, On sub- and superpramarts with values in a Banach lattice, Measure theory, Oberwolfach 1981 (Oberwolfach, 1981) Lecture Notes in Math., vol. 945, Springer, Berlin-New York, 1982, pp. 352–365. MR 675299
B. Faires and T. J. Morrison, The Jordan decomposition of vector-valued measures, Proc. Amer. Math. Soc. 60 (1976), 139–143. MR 419723, DOI 10.1090/S0002-9939-1976-0419723-X
D. H. Fremlin, Abstract Köthe spaces. II, III, Proc. Cambridge Philos. Soc. 63 (1967), 951–956; 957–962. MR 216272, DOI 10.1017/s0305004100041980
D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge University Press, London-New York, 1974. MR 0454575, DOI 10.1017/CBO9780511897207
Yukio K\B{o}mura and Shozo Koshi, Nuclear vector lattices, Math. Ann. 163 (1966), 105–110. MR 192327, DOI 10.1007/BF02052842
Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
Helmut H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. Third printing corrected. MR 0342978, DOI 10.1007/978-1-4684-9928-5
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039, DOI 10.1007/978-3-642-65970-6

Über Klassen majorisierbarer Operatoren auf Banachverbänden

Jerzy Szulga, On the submartingale characterization of Banach lattices isomorphic to $l_{1}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 1, 65–68 (English, with Russian summary). MR 478265
Bertram Walsh, Ordered vector sequence spaces and related classes of linear operators, Math. Ann. 206 (1973), 89–138. MR 328541, DOI 10.1007/BF01430979
Yau-Chuen Wong, Characterizations of the topology of uniform convergence on order-intervals, Hokkaido Math. J. 5 (1976), no. 2, 164–200. MR 454582, DOI 10.14492/hokmj/1381758667
Yau Chuen Wong and Kung Fu Ng, Nuclear spaces and generalized AL-spaces, Southeast Asian Bull. Math. 5 (1981), no. 1-2, 45–58. MR 638689