The Journal of the Indian Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

Pettis-Type Spaces for a Bounded Family of Measures


Affiliations
1 Department of Mathematics, The University of Burdwan, Burdwan-713104 (W.B.), India
     

   Subscribe/Renew Journal


Let (Ω,Σ,μ) be a probability space and let N⊂ca (Σ) be a bounded family of positive measures and X be a Banach space. Let P1(N,X) be the Pettis-type spaces with respect to N. Assuming that X is weakly sequentially complete, we prove the completeness of P1(N,X) with respect to the Pettis semi-norm. Also we prove the Vitali’s convergence theorem, Lebesgue dominated theorem and a necessary and sufficient condition for a function to belong to P1(N,X).

Keywords

Lebesgue-Type Spaces, Pettis-Type Spaces, Vitali’s Convergence Theorem, Lebesgue Dominated Convergence Theorem, Young’s Function, Orlicz Spaces.
Subscription Login to verify subscription
User
Notifications
Font Size


Abstract Views: 237

PDF Views: 0




  • Pettis-Type Spaces for a Bounded Family of Measures

Abstract Views: 237  |  PDF Views: 0

Authors

N. D. Chakraborty
Department of Mathematics, The University of Burdwan, Burdwan-713104 (W.B.), India
M. Sahu
Department of Mathematics, The University of Burdwan, Burdwan-713104 (W.B.), India
B. Sen
Department of Mathematics, The University of Burdwan, Burdwan-713104 (W.B.), India

Abstract


Let (Ω,Σ,μ) be a probability space and let N⊂ca (Σ) be a bounded family of positive measures and X be a Banach space. Let P1(N,X) be the Pettis-type spaces with respect to N. Assuming that X is weakly sequentially complete, we prove the completeness of P1(N,X) with respect to the Pettis semi-norm. Also we prove the Vitali’s convergence theorem, Lebesgue dominated theorem and a necessary and sufficient condition for a function to belong to P1(N,X).

Keywords


Lebesgue-Type Spaces, Pettis-Type Spaces, Vitali’s Convergence Theorem, Lebesgue Dominated Convergence Theorem, Young’s Function, Orlicz Spaces.