Let , be two probability measures on the same measurable space .

We draw jointly the couple of random variables according to the following procedure:

• Draw
• Draw a candidate and accept the candidate with probability with . Otherwise reject the candidate and set .

Obviously, where

We now show that is a coupling of . To do so, it is sufficient to check that for any bounded or non-negative function , .

Indeed, write:

The probability of coupling is given by:

Question: we know that . But I can't see how we can prove that