my question is half about the coupling, and half about the conditining so I wondered whether you worked with coupling
So I was thinking of the following: suppose I have two random variables $X,Y$ on a Polish space, where I know the distribution of $X$ being $\mu$ and the distribution of $Y$ given $X$, say $K(x,\mathrm dy)$
I don't have it in hand, but the point is that the maximal coupling is uniquely determined by the equality above, and by the fact that $$ \Bbb P(Z\in A,Z'\in A'|Z\neq Z') = \Bbb P(Z\in A|Z\neq Z')\cdot \Bbb P(Z'\in A'|Z\neq Z') $$
@MichaelGreinecker no, I was also surprised that he is from Holland :)
so I think that the following holds $$ \Bbb P(X = X') = 1-\frac12\|\mu - \mu'\| $$ and $$ \Bbb P(Y = Y'|X = X') = 1-\frac12\sup\limits_{x}\|K(x,\cdot) - K'(x,\cdot)\| $$
@MichaelGreinecker I think, $\Bbb P$ is constructed to give a positive mass to the diagonal
in the formula above for the conditional independence I use $Z = (X,Y)$ and $Z' = (X',Y')$
The maximal coupling of the distributions of $(X,Y)$ and $(X',Y')$ may not induce maximal couplings of $Y$ and $Y'$, which you seem to use when you let $\Bbb P(Y = Y'|X = X') = 1-\frac12\sup\limits_{x}\|K(x,\cdot) - K'(x,\cdot)\|$
