We study injective and projective tensor products of measurable Banach bundles. More precisely, given two separable measurable Banach bundles \(\textbf{E}\) , \(\textbf{F}\) defined over a probability space \((\textrm{X},\Sigma ,\mathfrak {m})\) , we construct two measurable Banach bundles \(\textbf{E}\hat{\otimes }_\varepsilon \textbf{F}\) and \(\textbf{E}\hat{\otimes }_\pi \textbf{F}\) over \((\textrm{X},\Sigma ,\mathfrak {m})\) , such that \(\Gamma (\textbf{E}\hat{\otimes }_\varepsilon \textbf{F})\cong \Gamma (\textbf{E})\hat{\otimes }_\varepsilon \Gamma (\textbf{F})\) and \(\Gamma (\textbf{E}\hat{\otimes }_\pi \textbf{F})\cong \Gamma (\textbf{E})\hat{\otimes }_\pi \Gamma (\textbf{F})\) , where \(\textbf{G}\mapsto \Gamma (\textbf{G})\) is the map assigning to a measurable Banach bundle \(\textbf{G}\) and its space of \(L^\infty (\mathfrak {m})\) -sections, while \(\Gamma (\textbf{E})\hat{\otimes }_\varepsilon \Gamma (\textbf{F})\) and \(\Gamma (\textbf{E})\hat{\otimes }_\pi \Gamma (\textbf{F})\) denote the injective and projective tensor products, respectively, of \(\Gamma (\textbf{E})\) and \(\Gamma (\textbf{F})\) in the sense of \(L^\infty (\mathfrak {m})\) -Banach \(L^\infty (\mathfrak {m})\) -modules. In combination with previous results, this provides a fiberwise representation of the injective tensor product \(\mathscr {M}\hat{\otimes }_\varepsilon \mathscr {N}\) and the projective tensor product \(\mathscr {M}\hat{\otimes }_\pi \mathscr {N}\) of two countably generated \(L^\infty (\mathfrak {m})\) -Banach \(L^\infty (\mathfrak {m})\) -modules \(\mathscr {M}\) , \(\mathscr {N}\) .
