Representations and couplings

Suppose $X$ are $Y$ are random elements of $X$, defined on the same probability space $(\Omega , \mathcal{F}, \mathbb {P})$, with distributions $P$ and $Q$. Show that $\mathbb {P}^{*}\left\{ X \neq Y\right\} \geq v(P, Q)$. Hint: Choose a measurable set $D \supseteq \left\{ X \neq Y\right\}$ with $\mathbb {P}\left(D\right) = \mathbb {P}^{*}\left\{ X \neq Y\right\}$. Note that $\mathbb {P}\left(X \in A\right)-\mathbb {P}\left(Y \in A\right) = \mathbb {P}\left(X \in A\right) \cap D-\mathbb {P}\left(Y \in A\right) \cap D$.

Suppose the diagonal $\Delta : = \left\{ (x, y) \in \mathcal{X} \times \mathcal{X}: x = y\right\}$ is product measurable. Recall from Section 3.3 that $v = 1-(P \wedge Q)(X) = (P-Q)^{+}(X) = (Q-P)^{+}(X)$. Define a probability measure $\mathbb {P} = \frac{1}{v}(P-Q)^{+} \otimes (Q-P)^{+}+\lambda$, where $\lambda$ is the image of $P \wedge Q$ under the map $x \mapsto (x, x)$. Let $X$ and $Y$ be the coordinate maps. Show that $X$ has distribution $P$ and $Y$ has distribution $Q$, and $\mathbb {P}\left(X \neq Y\right) = v$.

For each $x$ in $X$, show that there are at most countably many closed balls $B$ centered at $x$ with $P(\partial B) > 0$.

For each $x_{i}$ in $S$, find a ball $B_{i}$ centered at $x_{i}$ with radius between $\epsilon / 4$ and $\epsilon / 2$ and $P\left(\partial B_{i}\right) = 0$.

Show that $\cup_{i \in \mathbb {N}} B_{i}$ contains the closure of $S$. Hint: Each point of the closure lies within $\epsilon / 4$ of at least one $x_{i}$.

Show that $P\left(\cup_{i \leq m} B_{i}\right) > 1-\epsilon$ when $m$ is large enough.

Show that the sets $C_{i}: = B_{i} \backslash \cup_{1 \leq j < i} B_{j}$ and $C_{0}: = \left(\cup_{i \leq m} B_{i}\right)^{c}$ have the desired properties.

Suppose there exists a proper subset $A_{0}$ of $S$ for which $\nu A_{0} = \mu R_{A_{0}}$. Define $R_{\alpha }^{\prime } = R_{\alpha } \backslash R_{A_{0}}$ for $\alpha \notin A_{0}$. Show that $\nu A \leq \mu R_{A}^{\prime }$ for all $A \subseteq S \backslash A_{0}$. Construct $K$ by invoking the inductive hypothesis separately for $A_{0}$ and $S \backslash A_{0}$. (Compare with part (iv) of Problem [5].) Now suppose $\nu A < \mu R_{A}$ for all proper subsets $A$ of $S$. Write $L_{\alpha }$ for the probability distribution $\mu \left(\cdot \mid R_{\alpha }\right)$, which concentrates on $R_{\alpha }$.

Show that $\mu \geq \nu \left\{ 1\right\} L_{1}$. Hint: Show $\mu B \geq \nu \left\{ 1\right\} \mu \left(B R_{1}\right) / \mu R_{1}$ for all $B \subseteq R_{1}$.

Write $\epsilon_{1}$ for the unit mass at 1. Let $\theta_{0}$ be the largest value in $[0, \nu \left\{ 1\right\} ]$ for which $\left(\mu -\theta_{0} L_{1}\right) R_{A} \geq \left(\nu -\theta_{0} \epsilon_{1}\right) A$ for every $A \subseteq S$.

If $\theta_{0} = \nu \left\{ 1\right\}$, use the inductive hypothesis to find a probability kernel from $S \backslash \left\{ 1\right\}$ into $T$ for which $\left(\mu -\nu \left\{ 1\right\} L_{1}\right) \geq \sum_{\alpha \geq 2} \nu \left\{ \alpha \right\} K_{\alpha }$. Define $K_{1} = L_{1}$.

If $\theta_{0} < \nu (1)$, show that there exists an $A_{0} \subseteq S$ for which $\left(\mu -\theta L_{1}\right) R_{A_{0}} < \left(\nu -\theta \epsilon_{1}\right) A_{0}$ when $\nu \left\{ 1\right\} \geq \theta > \theta_{0}$. Deduce that $A_{0}$ must be a proper subset of $S$ for which $\left(\mu -\theta_{0} L_{1}\right) R_{A_{0}} = \left(\nu -\theta_{0} \epsilon_{1}\right) A_{0}$. Invoke part (i) to find a probability kernel $M$ for which $\mu -\theta_{0} L_{1} \geq \left(\nu \left\{ 1\right\} -\theta_{0}\right) M_{1}+\sum_{\alpha \geq 2} \nu \left\{ \alpha \right\} M_{\alpha }$. Define $K_{1}: = \left(\theta_{0} / v\left\{ 1\right\} \right) L_{1}+\left(1-\theta_{0} / v\left\{ 1\right\} \right) M_{1}$.