• Received : 2009.12.11
  • Published : 2011.07.31


In this paper we study the Banach space $L^1$(G) of real valued measurable functions which are integrable with respect to a vector measure G in the sense of D. R. Lewis. First, we investigate conditions for a scalarly integrable function f which guarantee $f{\in}L^1$(G). Next, we give a sufficient condition for a sequence to converge in $L^1$(G). Moreover, for two vector measures F and G with values in the same Banach space, when F can be written as the integral of a function $f{\in}L^1$(G), we show that certain properties of G are inherited to F; for instance, relative compactness or convexity of the range of vector measure. Finally, we give some examples of $L^1$(G) related to the approximation property.


Lebesgue space of vector measure;convergence in $L^1$(G);the range of vector measures;Lyapunov convexity theorem;the approximation property


  1. G. P. Curbera, Operators into $L^1$ of a vector measure and applications to Banach lattices, Math. Ann. 293 (1992), no. 2, 317-330.
  2. G. P. Curbera, When $L^1$ of a vector measure is an AL-space, Pacific J. Math. 162 (1994), no. 2, 287-303.
  3. G. P. Curbera, Banach space properties of $L^1$ of a vector measure, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3797-3806.
  4. J. Diestel, Sequences and series in Banach spaces, Springer-Verlag, New York, 1984.
  5. J. Diestel and J. J. Uhl, Jr., Vector measures, Amer. Math. Soc. Surveys Vol. 15, Providence, Rhode. Island, 1977.
  6. G. Knowles, Lyapunov vector measures, SIAM J. Control 13 (1975), 294-303.
  7. D. R. Lewis, Integration with respect to vector measures, Pacific J. Math. 33 (1970), 157-165.
  8. D. R. Lewis, On integrability and summability in vector spaces, Illinois. J. Math. 16 (1972), 294-307.
  9. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Spaces, Springer, Berlin, 1977.
  10. A. Lyapunov, Sur les fonctions-vecteurs completement additivies, Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 465-478.
  11. P. Meyer-Nieberg, Banach Lattices, Springer, Berlin and New york, 1991.
  12. S. Okada, W. J. Ricker, and L. Rodriguez-Piazza, Compactness of the integration operator associated with a vector measure, Studia. Math. 150 (2002), no. 2, 133-149.
  13. V. I. Rybakov, On the theorem of Bartle, Dunford and Schwartz concerning vector measures, Mat. Zametki 7 (1970), 247-254.
  14. G. F. Stefansson, $L_1$ of a vector measure, Matematiche (Catania) 48 (1993), 219-234.
  15. A. Szankowski, A Banach lattice without the approximation property, Israel J. Math. 24 (1976), no. 3-4, 329-337.