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A b s t r a c t . The tensor integral of a vector valued function f : Ω → X with respect to a countably additive vector valued measure ν : Σ → Y has been defined by Stefansson in [14] and he has investigated many of its properties. The integral is an element of the injective tensor product X⊗ Y . We study the Banach space L1(ν, X, Y ) of all⊗-integrable functions and discuss many properties of this space. We also study the space w-L1(ν, X, Y ) of all weakly⊗-integrable functions.

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