On ultra-products of some families of composition operators between certain finite dimensional ℓp spaces

Let 0 < σ < 1 and 1 < p, r < ∞ be such that 1/r + (1- σ)/p' = 1. We show that for every continuous linear map T between Banach spaces E, F such that its restriction to every finite dimensional subspace N of E factorizes through a chain of type

ℓ^∞ (Ω_N,μ_N) → ℓ^{r  ∞} (Ω_N,μ_N) →  ℓ ¹ (Ω_N,μ_N)+ ℓ^p((Ω_N,μ_N)

where (Ω_N,μ_N) is a discrete measure space with a finite number of atoms, there

is a σ- finite measure space (Ω_, μ) such that T ∈ L(E, F") factorizes through the chain of "continuous spaces

ℓ^∞ (Ω_N,μ_N) → ℓ^{r  ∞} (Ω_N,μ_N) →  ℓ ¹ (Ω_N,μ_N)+ ℓ^p((Ω_N,μ_N)