Pinsker's Inequality for Metric Spaces!

Question

Let $p$ and $q$ be two probability distributions on a countable set $X$. Then the total variation distance $V(p,q)$ between $p$ and $q$ is defined as follows: \begin{equation} V(p,q)=\frac{1}{2}\sum_{x\in X}|p(x)-q(x)| \end{equation}

Also, Pinkser's Inequality relates the total variation distance to the KL-divergence as follows: \begin{equation*} V(p,q)\leq\sqrt{\frac{1}{2}D_{KL}(p\|q)} \end{equation*}

I am trying to generalize this result to compact metric spaces i.e., Let $X$ be a compact metric spaces that can be partitioned into $K$ disjoint sets denoted by $C_k$, $k=1,2,\ldots,K$ such that \begin{align*} \cup_{k=1}^KC_k=X, &&\text{and}\quad C_i\cap C_j=\emptyset \quad i\ne j \end{align*}

Can I do it directly by defining $V(p,q)$ as follows: \begin{equation} V(p,q)=\frac{1}{2}\sum_{i=1}^K|p(C_i)-q(C_i)| \end{equation}

and does the Pinkser inequality still hold?

Answer 1

Yes.

Just define a distribution $p'$ on the set $Y = \{C_1, \ldots, C_K\}$ by $P(p' =C_k) = P(p \in C_k)$ for each $k \le K$. Then apply the countable set case.