The existence of probability measures with specified projections

V. N. Sudakov

Matematičeskie zametki, Tome 14 (1973) no. 4, pp. 573-576.

Résumé

Let $X$ and $Y$ be locally compact $\sigma$-compact topological spaces, $F\subset X\times Y$ is closed, and $P(F)$ is the set of all Borel probability measures on $F$. For us to find, for the pair of probability measures $(\mu_X,\mu_Y)\in P(X)\times P(Y)$, a probability measure $\mu\in P(F)$ such that $\mu_X=\mu\pi_X^{-1}$, $\mu_Y=\mu\pi_Y{-1}$ it is necessary and sufficient that, for any pair of Borel sets $A\in X$, $B\subset Y$ for which $(A\times B)\cap F=\emptyset$, the condition $\mu_XA+\mu_YB\le1$ holds.