A set $C \in \mathcal{B}(\mathrm{X})$ is called a small set if there exists an $m>0$, and a non-trivial measure $\nu_m$ on $\mathcal{B}(\mathrm{X})$, such that for all $x \in C, B \in \mathcal{B}(\mathrm{X})$,
$$ P^m(x, B) \geq \nu_m(B) $$
When this holds we say that $C$ is $\nu_m$-small.
(from 5.2 in \cite{meyn2012markov})
An irreducible chain on a countable space $X$ is called (i) aperiodic, if $d(x) \equiv 1, x \in \mathrm{X}$; (ii) strongly aperiodic, if $P(x, x)>0$ for some $x \in \mathrm{X}$.
(from 5.4.2 in \cite{meyn2012markov})
We will call a set $C \in \mathcal{B}(\mathrm{X}) \nu_a$-petite if the sampled chain satisfies the bound
$$ K_a(x, B) \geq \nu_a(B) $$
for all $x \in C, B \in \mathcal{B}(\mathrm{X})$, where $\nu_a$ is a non-trivial measure on $\mathcal{B}(\mathrm{X})$. (from 5.5.2 in \cite{meyn2012markov})
(Proposition 4.2.1. \cite{meyn2012markov}) The following are equivalent formulations of $\varphi$-irreducibility: (i) for all $x \in \mathrm{X}$, whenever $\varphi(A)>0, U(x, A)>0$; (ii) for all $x \in \mathrm{X}$, whenever $\varphi(A)>0$, there exists some $n>0$, possibly depending on both $A$ and $x$, such that $P^n(x, A)>0$; (iii) for all $x \in \mathrm{X}$, whenever $\varphi(A)>0$ then $K_{a_{\frac{1}{2}}}(x, A)>0$.
$\psi$-Irreducibility notation (i) The Markov chain is called $\psi$-irreducible if it is $\varphi$-irreducible for some $\varphi$ and the measure $\psi$ is a maximal irreducibility measure satisfying the conditions of Proposition 4.2.2. (ii) We write
$$ \mathcal{B}^{+}(\mathrm{X}):={A \in \mathcal{B}(\mathrm{X}): \psi(A)>0} $$
for the sets of positive $\psi$-measure; the equivalence of maximal irreducibility measures means that $\mathcal{B}^{+}(\mathrm{X})$ is uniquely defined. (iii) We call a set $A \in \mathcal{B}(\mathrm{X})$ full if $\psi\left(A^c\right)=0$. (iv) We call a set $A \in \mathcal{B}(\mathrm{X})$ absorbing if $P(x, A)=1$ for $x \in A$.
we will get $\mu^{\mathrm{Leb}}$-irreducibility, where $\mu^{\mathrm{Leb}}$ is Lebesgue measure
more details of mcmc theory that is related see \cite{roberts1998markov}
The state space is denoted by $\mathcal{X} \subseteq \mathbb{R}^k$, equipped with the Borel $\sigma$-algebra $\mathscr{B}$. All distributions, including $\pi(x)$ and transition densities$q(x, y)$, are defined with respect to the Lebesgue measure $\mu^{\text{Leb}}$.
Proposal Kernel: A candidate transition is proposed from $x \in \mathcal{X}$ to $y \in \mathcal{X}$ according to a proposal density $q(x, y)$, where $q(x, \cdot)$ is the conditional density for generating proposals given the current state $x$.
The actual transition kernel $P(x, A)$, which governs the evolution of the Markov chain, is defined by: $$ P(x, A) = \int_A p(x, y) , \mathrm{d}y + \delta_x(A) r(x), $$ where: -$p(x, y) = q(x, y) \alpha(x, y)$ is the transition density for $y \neq x$, -$r(x) = \int q(x, y) [1 - \alpha(x, y)] , \mathrm{d}y$ is the probability of remaining at $x$, -and $\delta_x(A)$ is the Dirac delta measure, assigning all mass to the point $x$.
Stationary Distribution: The Markov chain is designed to satisfy detailed balance, ensuring that $\pi(x)$ is invariant: $$ \pi(A) = \int_\mathcal{X} \pi(x) P(x, A) , \mathrm{d}x, \quad \forall A \in \mathscr{B}. $$
Convergence in Distribution: Let $P^n(x, A)$ denote the $n$-step transition probabilities of the Markov chain, defined as: $$ P^n(x, A) = P(\Phi_n \in A \mid \Phi_0 = x), \quad x \in \mathcal{X}, A \in \mathscr{B}. $$ Under suitable irreducibility and aperiodicity conditions, it holds that: $$ |P^n(x, \cdot) - \pi|{\text{TV}} \to 0 \quad \text{as } n \to \infty, $$ for $\pi$-almost (holds outside every null set) all $x$. Here, the total variation distance is defined as: $$ |P^n(x, \cdot) - \pi|{\text{TV}} := \frac{1}{2} \sup_{A \in \mathscr{B}} \left| P^n(x, A) - \pi(A) \right|. $$