Tensor products of measurable Banach bundles

We study injective and projective tensor products of measurable Banach bundles. More precisely, given two separable measurable Banach bundles E E , F F defined over a probability space (X,Σ ,m) ( X , Σ , m ) , we construct two measurable Banach bundles E⊗ ̂_ε F E ⊗ ^ ε F and E⊗ ̂_π F E ⊗ ^ π F over (X,Σ ,m) ( X , Σ , m ) , such that Γ (E⊗ ̂_ε F) Γ (E)⊗ ̂_ε Γ (F) Γ ( E ⊗ ^ ε F ) ≅ Γ ( E ) ⊗ ^ ε Γ ( F ) and Γ (E⊗ ̂_π F) Γ (E)⊗ ̂_π Γ (F) Γ ( E ⊗ ^ π F ) ≅ Γ ( E ) ⊗ ^ π Γ ( F ) , where G↦ Γ (G) G ↦ Γ ( G ) is the map assigning to a measurable Banach bundle G G </jats:alt