Let $f(x):[0,+\infty)\to [0,+\infty)$ be a function such that $f(x)\ge x$ almost sure. Consider the iteration $x_{n+1}=f(x_n)$, do we have the set $A=\{x_0>0 | \lim_\limits{n\to\infty} x_n=0\}$ has measure zero?
Here we add an assumption that $\mu(f(A))= 0$ if and only if $\mu(A)=0$.
$\{x_n\}$ is an increasing sequence for almost all $x_0$: $\{x_0>0: x_{n+1} < x_n\}$ has measure $0$ for each $n$ and their union also has measure $0$. If $x_0$ is such that $\{x_n\}$ is an increasing then $x_0$ cannot belong to $A$. Hence $A$ has measure $0$.