Abstract: Let (Ω, ∑,μ) be a measure space and (X_ω)_ω∈Ω a family of closed linear subspaces of some Banach space X. Let L₁ (μ, (X_ω)) be the linear subspace of the Bochner space L₁ (μ,X) given by L₁ (μ, (X_ω)):={f ∈ L₁ (μ,X):f(ω) ∈ X_ω for all ω ∈ Ω}. Suppose that each X_ω is of type p with constant 1 for some fixed p > 1. Two results are obtained:(1) If E ⊆ L₁ (μ, (X_ω)) is isomorphic to some L₁ space with constant less than 2^1-1/p, then there is a projection from L₁ (μ,X) onto E. (2) If E ⊆ L₁ (μ, (X_ω)) is a L_1,λ space with λ small enough then E is complemented in L₁ (μ, (X_ω)). These results generalize early results of Dor[3], Alspach and Johnson[1].