Definition (Total Variation Metric)

For two Probability Measures $\mu,\nu \in\mathcal{P}(\mathbb{X})$, the total Variation metric is given by $$\begin{align*} \lVert \mu-\nu \rVert _{TV} :&= 2\sup_{B\in\mathcal{B}(\mathbb{X})}\lvert\mu(B)-\nu(B)\rvert\\ &=\sup_{f:\lVert f \rVert _{\infty}\le 1}\left\lvert \int\limits f(x) \, \mu(dx)-\int\limits f(x) \, \nu(dx) \right\rvert \end{align*}$$ where the Supremum is over all measurable real $f$ s.t. $\lVert f \rVert_{\infty}=\sup_{x\in\mathbb{X}}\lvert f(x)\rvert\le 1$.

Definition (Converge in Total Variation)

Let $(\mu_{n})_{n\in\mathbb{N}}\subset \mathcal{P}(\mathbb{X})$ be a sequence of Probability Measures in the space of probability measures. $\mu_{n}\to \mu$ in total variation if $$\lVert \mu_{n}-\mu \rVert _{TV}\to0$$or $$2\sup_{B\in\mathcal{B}(\mathbb{X})}\left| \mu_{n}(B)-\mu(B) \right| \to0$$

Definition (Continuous in Total Variation)

Let $\mathbb{X}$ be standard Borel and let $P(\cdot\mid \cdot )\in\mathcal{P}(\mathbb{X}\mid \mathbb{X})$ be a Stochastic Kernel. We say $P$ is continuous in total variation if and only if for any $x \in\mathbb{X}$ then $\forall (x_{n})_{n\in\mathbb{N}}\subset \mathbb{X}$ s.t. $x_{n}\to x$ we have that $$P(\cdot\mid x_{n})\to P(\cdot\mid x)\text{ in total variation}$$