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A coupling of two probability measures $\mathbb{P}$ and $\mathbb{P}'$ on the same measurable space $(E,\mathcal{E})$ is a probability measure $\tilde{\mathbb{P}}$ on $(E\times E,\mathcal{E}\otimes\mathcal{E})$ with $\mathbb{P}=\tilde{\mathbb{P}}\circ\pi^{-1}$ and $\mathbb{P}'=\tilde{\mathbb{P}}\circ\pi'^{-1}$, where $\pi(x,x')=x, \pi'(x,x')=x'$ for $(x,x')\in E\times E$.

My question is: Is there ALWAYS a coupling for any two probability measures on the same measurable space?

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    $\begingroup$ Yes--define $\tilde P(A\times B)=P(A)P'(B)$ for every $A$ and $B$ in $\mathcal E$ and extend $\tilde P$ to $\mathcal E\otimes\mathcal E$. $\endgroup$
    – Did
    Dec 31, 2014 at 14:59

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Yes. You can choose two independent random variables $X, Y$ distributed according to $\mathbb{P}$ and $\mathbb{P}'$ respectively.

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