The Journal of the Indian Mathematical Society

N. D. Chakraborty ¹, M. Sahu ¹, B. Sen ¹

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

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.

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