Title: A note on relative amenable of finite von Neumann algebras,
Abstract: Let $M$ be a finite von Neumann algebra (resp. a type II$_{1}$ factor) and
let $N\subset M$ be a II$_{1}$ factor (resp. $N\subset M$ have an atomic part).
We prove that the inclusion $N\subset M$ is amenable implies the identity map
on $M$ has an approximate factorization through $M_m(\mathbb{C})\otimes N $ via
trace preserving normal unital completely positive maps, which is a
generalization of a result of Haagerup. We also prove two permanence properties
for amenable inclusions. One is weak Haagerup property, the other is weak
exactness.