TAILIEUCHUNG - Inclusions and the approximate identities of the generalized grand Lebesgue spaces
Let (Ω,∑, µ) and (Ω,∑, v) be two finite measure spaces and let Lp),θ (µ) and Lq),θ (v) be two generalized grand Lebesgue spaces, where 1 0 such that ∥f ∥q),θ,υ ≤ C (p, q) ∥f ∥p),θ,µ (2) for all f ∈ Lp),θ (µ) . Proof Assume that the inequality (2) is satisfied and µ ≈ υ . By the inequality (2) the inclusion Lp),θ (µ) ⊆ Lq),θ (υ) holds in the sense of individual functions. Then by Lemma 1, the inclusion Lp),θ (µ) ⊆ Lq),θ (υ) holds in the sense of equivalence classes. Conversely, assume that Lp),θ (µ) ⊆ Lq),θ (υ) holds in the sense of equivalence classes. The grand Lebesgue space Lp),θ (µ) is a Banach space with the sum norm ∥f ∥ = ∥f ∥p),θ,µ + ∥f ∥q),θ,υ . ( ) Indeed, if we get any Cauchy sequence (fn )n∈N in the normed space Lp),θ (µ) , ∥.∥ , it is also a Cauchy sequence ( ) ) ( in the spaces Lp),θ (µ) , ∥.∥p),θ,µ and Lq),θ (υ) , ∥.∥q),θ,υ . Then (fn )n∈N converges to functions f and g in spaces Lp),θ (µ) and Lq),θ (v) , respectively . Thus, one can find a subsequence (fni ) of (fn ) such that fni → f, µ − and fni → g, υ − . Since v is absolutely continuous .
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