Probability measures are tight on the product space if and only if each marginal distribution is tight

Question

Probability measures on $S' \times S'' $ are tight if and only if the two sets of marginal distributions are tight on $S'$ and $S''$.

I can prove the only if direction as follows. By tightness, we mean, for every $\epsilon >0$ there is a compact set $K$ such that $P(K) > 1-\epsilon$, where $P$ is a measure on the product space. Then, the take $K' = \pi_1(K)$, which is compact by continuity of the projection map, has $P_1(K') = P(K' \times S'')\ge P(K)>1-\epsilon.$ So we have tightness for the marginal distributions.

However, I cannot prove the if direction. I would greatly appreciate any help.

Answer 1

Use: $(K_1 \times K_2)^{c} \subset (K_1 \times S'')^{c} \cup (S' \times K_2)^{c})$. Let me know if you need more details.