The Lubell bound for intersecting-union families

In a 2021 survey on Katona's circle method, Frankl conjectured that every family $\mathcal{F}\subseteq 2^{[n]}$ in which any two members intersect and no two members cover $[n]$ satisfies the sharp Lubell-type bound $\sum_{F\in \mathcal{F}}\binom{n}{|F|}^{-1}\le \frac{n+1}{6}.$ This improves the earlier estimate $\frac{n}{4}$ obtained by the circle method. In this paper, we prove Frankl's conjecture and determine all extremal families. Our proof replaces the cyclic permutation argument with a $p$-biased measure framework on the Boolean lattice, and then integrates the resulting estimates over the full probability range. This continuous integration recovers the optimal coefficient $\frac{1}{6}$, whereas the discrete averaging inherent in the circle method yields only $\frac{n}{4}$.