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Pettis-Type Spaces for a Bounded Family of Measures
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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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