  In this article, we prove Neveu decomposition for the action of the locally
compact amenable semigroup of positive contractions on semifinite von Neumann
algebras and thus, it entirely resolves the problem for the actions of
arbitrary amenable semigroup on semifinite von Neumann algebras. We also prove
it for amenable group actions by Markov automorphisms on any $\sigma$-finite
von Neumann algebras. As an application, we obtain stochastic ergodic theorem
for actions of $ \mathbb{Z}_+^d$ and $\mathbb{R}_+^d$ for $ d \in \mathbb{N}$
by positive contractions on $L^1$-spaces associated with a finite von Neumann
algebra. It yields the first ergodic theorem for positive contraction on
non-commutative $L^1$-spaces beyond the Danford-Schwartz category.
