A note on characterizations of relative amenability on finite von Neumann algebras

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.