Let X be a Banach space and (f
n
)
n
be a bounded sequence in L
1(X). We prove a complemented version of the celebrated Talagrand's dichotomy, i.e., we show that if (en
)
n
denotes the unit vector basis of c
0, there exists a sequence gn
∈ conv(f
n
, f
n+1,...) such that for almost every ω, either the sequence (gn
(ω) ⊗ en
) is weakly Cauchy in or it is equivalent to the unit vector basis of ℓ
1. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of ℓ
1 in L
1(X). As an application, we show that for a Banach space X, the space L
1(X) has Pełczyńiski's property (V*) if and only if X does.