For a complete measure space ( X , ∑ , μ ) , we give conditions which force L p ( X , μ ) , for 1 ≤ p < ∞ , to be isometrically isomorphic to ℓ p ( Γ ) for some index set Γ which depends only on ( X , μ ) . Also, we give some new characterizations which yield the inclusion L p ( X , μ ) ⊂ L q ( X , μ ) for 0 < p < q .