The Journal of the Indian Mathematical Society

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Pettis-Type Spaces for a Bounded Family of Measures


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

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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.
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  • Pettis-Type Spaces for a Bounded Family of Measures

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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.