Let $X = \prod \_{s \in S} \Omega\_s$, with $\Omega\_s$ finite and all the same, $S$ countable. Let $\mu\_1$ and $\mu\_2$ be two probability measures on the product space (not necessarily the product measure). Let $C$ be a coupling between the two measures and let's define,


$$
rift(C) = sup\_{s \in S}\lbrace \mbox{ } C \mbox{ } \mbox{ } \lbrace ( \omega^1, \omega^2 ) \in X \times X \ \text{ s.t. } \omega^1\_s \neq \omega^2\_s \mbox{ } \rbrace \mbox{ } \mbox{ } \rbrace.$$


Then we call distance between the two measures $dist(\mu\_1, \mu\_2)$ the infimum over all the possible couplings $c$ of the previous quantity. 


How can I prove that, if A is a cylinder subset of $X$, specified by $r$ components $\omega\_i$, then $\forall \mu\_1, \mu\_2$ probability measures as before, $$|\mu\_1(A) - \mu\_2(A)| \leq r \mbox{ } dist(\mu\_1, \mu\_2)$$


Definition: the coupling is not necessarily the product between the two measures. Let's consider two measures $\mu^1$ and $\mu^2$, acting each one on the $\sigma$-algebra of subsets of $X$. The coupling $C$ is any measure acting on the $\sigma$-algebra of subsets of $X^1 \times X^2$, with $X^i = X$, which has as marginals $\mu^1$ and $\mu^2$, i.e. $\mu^1 = C \circ \pi^{-1}\_1$, $\mu^2 = C \circ \pi^{-1}\_2$, where $\pi\_i$ is the projection $\pi\_i : X^1 \times X^2 \longrightarrow X^i$.



inequality for coupling of measures
Here is what seems to me a good beginning, if the argument can be carried all the way. 


We have that $dist(\mu\_1,\mu\_2):=\inf\_c \sup\_s f(s,c)$,


where $f(s,c):= c( \{ (\omega^1,\omega^2) \in X\times X : \omega^1\_s \neq \omega^2\_s \} )$.


Then for any indices $ \{ s\_i \}$ for $i$ from $1$ to $r$, we have


$\sup\_s f(s,c) \geq f(s\_i,c)$, so that


$r \sup\_s f(s,c) \geq \sum\_{i=1}^r f(s\_i,c)$,


and taking the infimum in $c$ we have


$r dist(\mu\_1,\mu\_2) \geq \inf\_c \sum\_{i=1}^r f(s\_i,c)$.


But $c$ is a measure, and so we should be able to arrange the right hand side as $c$ of some set where the components differ in at most $r$ spots, the infimum of which can be bounded below by the difference of the measures. 


For this to work, maybe the set in the definition of $rift$ should have ONLY $\omega^1\_s \neq \omega^2\_s$?


