  We prove the invariance of the Gibbs measure for the periodic
Schrodinger-Benjamin-Ono system (when the coupling parameter |\gamma| \ne 0, 1)
by establishing a new local well-posedness in a modified Sobolev space and
constructing the Gibbs measure (which is in the sub-L^2 setting for the
Benjamin-Ono part.) We also show the ill-posedness result in H^s(\mathbb{T})
\times H^{s-{1/2}}(\mathbb{T}) for s < {1/2} when |\gamma| \ne 0, 1 and for any
s \in \mathbb{R} when |\gamma| =1.
