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Let $X, Y$ be second-countable topological spaces. Then the Borel $\sigma$-algebra of $X \times Y$ coincides with the product of those of $X,Y$. Let $\mu, \nu$ be Borel probability measures (b.p.m.) on $X,Y$ respectively. Let $\pi \in \Pi(\mu, \nu)$, i.e., $\pi$ is a b.p.m on $X\times Y$ with marginals $\mu$ on $X$ and $\nu$ on $Y$. Let $X',Y'$ be Borel subsets of $X,Y$ respectively such that $\pi( X' \times Y')>0$. We define a b.p.m. $\pi'$ on $X'\times Y'$ by $$ \pi'(B) := \frac{\pi(B)}{\pi(X' \times Y')} $$ for all Borel subset $B$ of $X' \times Y'$. Let $\mu' := P_\sharp^{X'} \pi'$ and $\nu' := P_\sharp^{Y'} \pi'$ be the marginals of $\pi'$ on $X'$ and $Y'$ respectively. Here $P^{X'}, P^{Y'}$ are the projection maps. Let $\pi^* \in \Pi(\mu', \nu')$. We define a b.p.m. on $X \times Y$ by $$ \overline \pi (B) := \pi(X' \times Y') \pi^*(B \cap (X'\times Y')) + \pi(B \cap (X'\times Y')^c) $$ for all Borel subset $B$ of $X \times Y$.
Then $\overline \pi \in \Pi(\mu, \nu)$.
For all Borel subset $B$ of $X$, we have \begin{align} \overline \pi (B \times Y) &= \pi (X' \times Y') \pi^* ((B \times Y) \cap (X' \times Y')) + \pi ((B \times Y) \cap (X' \times Y')^c) \\ &= \pi (X' \times Y') \pi^* ((B \cap X') \times Y') + \pi ((B \times Y) \cap (X' \times Y')^c) \quad (\star) \\ &= \pi (X' \times Y') \mu' (B \cap X') + \pi ((B \times Y) \cap (X' \times Y')^c) \\ &= \pi (X' \times Y') \pi' ((B \cap X') \times Y') + \pi ((B \times Y) \cap (X' \times Y')^c) \\ &= \pi (X' \times Y') \frac{\pi ((B \cap X') \times Y')}{\pi (X' \times Y')} + \pi ((B \times Y) \cap (X' \times Y')^c) \\ &= \pi ((B \cap X') \times Y') + \pi ((B \times Y) \cap (X' \times Y')^c) \\ &= \pi ((B \times Y) \cap (X' \times Y')) + \pi ((B \times Y) \cap (X' \times Y')^c) \quad (\star\star) \\ &= \pi (B \times Y) \\ &=\mu(B). \end{align}
In $(\star)$ and $(\star\star)$, we use $(B \times Y) \cap (X' \times Y') = (B \cap X') \times Y'$.