Definition (Petite Set)

A set $A\in\mathcal{B}(\mathbb{X})$ is called a petite set if for some probability measure $\mathcal{T}$ on $\mathbb{N}$ $$\sum_{n=0}^{\infty}\mathcal{T}(n)P_{xB}^{n}\ge\nu(B), \ \ \forall B\in\mathcal{B}(\mathbb{X})\text{ and }\forall x\in A$$for some positive measure $\nu$ also we can take $$\mathcal{T}(n)=(1-\epsilon)^{n-1}\epsilon, \ \epsilon\in(0,1)$$