Let $S$ be a petite set, $b\in\mathbb{R}$, $\epsilon>0$,and $V:\mathbb{X}\to \mathbb{R}^{+}$, Let $\{ X_{k} \}_{k\in\mathbb{N}}$ be a Irreducible MC. If the following is satisfied $\forall x\in\mathbb{X}$ $$E[V(x_{k+1})|x_{k}=x]\le V(x)-\epsilon+b\mathbb{1}_{\{ x\in S \}}$$ then $\{ X_{k} \}_{k\in\mathbb{N}}$ is Positive Harris Recurrent (equivalently $\exists$ an Invariant probability measure $\pi$).

Let $S$ be a petite set, $b\in\mathbb{R}^{+}$, and $V:\mathbb{X}\to \mathbb{R}^{+}$, $f:\mathbb{X}\to[0,\infty)$ for some $\epsilon>0$. Let $\{ x_{k} \}_{k\in\mathbb{N}}$ be a Markov chain. If the following holds $$E[V(x_{k+1})|x_{k}=x]\le V(x)-\epsilon+b, \ \ \ \ x\in\mathbb{X}$$then for any Invariant probability measure $\pi$ we have finite expectation $$\int\limits f(x) \, \pi(dx)\le b$$

Let $S$ be a compact set, $b<\infty$, and $V:\mathbb{X}\to \mathbb{R}^{+}$ s.t. $$\forall\alpha\in\mathbb{R}^{+}, \ \{ x:V(x)\le \alpha \}$$ is compact or equivalently $$\lim_{ \|x\| \to \infty } V(x)=\infty$$Let $\{ x_{k} \}_{k\in\mathbb{N}}$ be a Markov chain. Furthermore, let $\tau_{S}=\min\{ t>0: x_{t}\in S \}$, $\tau_{B_{N}}=\{ t>0:x_{t}\in B_{N} \}$ where $B_{N}=\{ z:V(z)\ge N \}$, if we have that $$P_{x}(\min(\tau_{S},\tau_{B_{N}})<\infty)=1$$then $$\forall x\in\mathbb{X}, \ \ E[V(x_{k+1})|x_{k}=x]\le V(x)+b\mathbb{1}_{\{ x\in S \}}\implies P(\tau_{S}<\infty)=1$$