On the full range of Zippin and inclusion indices of rearrangement-invariant spaces

Let X be a rearrangement-invariant space on [0, 1]. It is known that its Zippin indices β̲ _X,β¯ _X and its inclusion indices γ _X,δ _X are related as follows: 0≤β̲ _X≤1/γ _X≤1/δ _X≤β¯ _X≤1. We show that given β̲,β¯∈[0,1] and γ,δ∈[1,∞] satisfying β̲≤1/γ≤1/δ≤β¯, there exists a rearrangement-invariant space X such that β̲ _X=β̲, β¯ _X=β¯ and γ _X=γ, δ _X=δ.