Let $\pi$ be a coupling of probability measures $\mu,\nu.$ For measurable sets $A,B$ we have $$ \pi(A\times B) \leq \mu(A) $$ and $$ \pi(A^c \times B^c) \leq \mu(A^c), $$ Therefore we have $$ \pi(A\times B) = \mu(A). $$ Is this argument correct?
2026-02-22 23:25:30.1771802730
Question about Coupling of Measures
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