  Let $X$ be a Banach space with a basis $(e_k)_k$ and biorthogonals
$(e^\ast_k)_k$. An operator on $X$ is said to have a $\textit {large diagonal}$
if $\inf\limits_{k} |e_k^\ast(T(e_k))| > 0$. The basis $(e_k)_k$ is said to
have the $\textit {factorization property}$ if the identity factors through any
operator with a large diagonal. Under the assumption that the Rademacher
sequence is weakly null, we study the factorization property of the Haar system
in a Haar system space. A Haar system space is the completion of the span of
characteristic functions of dyadic intervals with respect to a rearrangement
invariant norm. We show that every bounded operator with a large diagonal on a
Haar system space is approximatively a factor of some diagonal operator with a
large diagonal. Moreover, when the Haar system is an unconditional basis for a
Haar system space, it has the factorization property.
