The book "Markov chain and stochastic stability" states the following proposition:
The discrete-time random walk $\Phi=\{\Phi_n\}$ on the half line $[0,\infty)$ with increment variable $W$ is $\varphi$-irreducible, with $\varphi((0,\infty))=0$ and $\varphi(\{0\})=1$, if and only if $P(W<0)>0$.
Obviously, this random walk is not $\varphi$-irreducible (never return to $0$) if $P(W<0)=0$, but I am wondering whether we can define other $\varphi'$ such that the random walk is $\varphi'$-irreducible given $P(W<0)=0$. For example, can we define $\varphi'$ such that $\varphi'(X)=1$ if $[10,\infty)\subseteq X$ and $\varphi'(X)=0$ if $[10,\infty)\not\subseteq X$ on the $\sigma-$algebra generated by a countable sequence $[10,\infty)\subset A_1\subset A_2\subset\cdots$?