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Source. Published p. 272, Theorem 3.1 (PDF).

Statement. If $0<p\le1/2$, $0<\beta<1$, and every cross intersection of $\mathcal F,\mathcal G$ has size greater than $\beta n$, then

$$\mu_p(\mathcal F)\mu_p(\mathcal G) \le 2^{2n(H((1+\beta)/2)-1)}.$$

The source additionally assumes $\beta<p$; the proof gives the displayed extension throughout $0<\beta<1$.

Proof. Replace each family by its upward closure. Cross intersections can only grow, and both measures can only increase. For an up-set $\mathcal U$, $\mu_p(\mathcal U)$ is nondecreasing in $p$: couple all coordinates using independent uniform variables $U_i\in[0,1]$ and take $\{i:U_i\le p\}$. Increasing $p$ only adds elements, so membership in an up-set cannot be lost. Thus each measure is at most its value at $p=1/2$. Apply theorem_2_1 and divide its product bound by $4^n$. $\square$

Source precision. The closing self-reference to Theorem 3.1 in its own proof on p. 272 is a reference to the unweighted Theorem 2.1.

Dependencies. theorem_2_1.