Let $i\in\{1,2\}$. The Measure Product Theorem states that, given the measure spaces $(X_i,\Sigma_i,\mu_i)$, there is at least one product measure $\pi$ such that $\pi(A_1\times A_2)=\mu_1(A_1)\;\mu_2(A_2)$, for $A_i\in\Sigma_i$.
It also states that if the $\mu_i$'s are $\sigma$-finite, then $\pi$ is unique.
I'd like an example of a non-$\sigma$-finite pair of measures from which, nevertheless, we only obtain one product, thus showing "if" cannot be "iff" on the paragraph above.