  In this paper, we give another two characterizations of relative amenability
on finite von Neumann algebras, one of which can be thought of as an analogue
of injective operator systems. As an application, we prove a stable property of
relative amenable inclusions. We prove that under certain assumptions, the
inclusion $N=\int_{X} \bigoplus N_{p} d \mu\subset M=\int_{X} \bigoplus M_{p} d
\mu$ is amenable if and only if $N_p\subset M_p$ is amenable almost everywhere.
