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If $c$ is a coupling between two measures, $c= \mu^1\, \, t \, \, \mu^2$, ($t$ is the symbol of binary operator of the coupling (I can't find a more proper symbol here)). The measures are both defined on a product space $\Omega = \prod_{s \in S} \Omega_s$. how is it justifiable writing that,

$(\mu^1 \circ \pi^{-1}_s)\, \, t \, \, (\mu^2 \circ \pi_s^{-1} ) \, \, \, = \, \, \, (\mu^1 \, t \, \, \mu^2)\, \, \circ \, \pi_s^{-1}\, \,\,$ tha means that the projection of the coupling is the coupling of the projections?

P.S. the coupling is not necessarily the product between the measures.

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    How do you know that the identity holds?2012-11-24

1 Answers 1

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You have some serious problems of definitions. First, the coupling of two probability measures being non unique (except in some degenerate cases), the notation $\mu_1 \ t\ \mu_2$ is quite unfortunate. Second, $c$ being a measure on $\Omega\times\Omega$ (and not $\Omega$), the object $(\mu_1\ t\ \mu_2)\circ\pi_s^{-1}$ is undefined.

Once this conundrum is solved, I believe the identity you are in fact trying to prove will become obvious if you translate your hypothesis on couplings in terms of random variables defined on some common probability space.