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

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.

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.