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The union-closed sets conjecture, attributed to P\'eter Frankl from 1979,states that for any non-empty finite union-closed family of finite sets notconsisting of only the empty set, there is an element that is in at least halfof the sets in the family. We prove a version of Frankl's conjecture forfamilies distributed according to any one of infinitely many distributions. Asa corollary, in the intersection-closed reformulation of Frankl's conjecture,we obtain that it is true for families distributed according to any one ofinfinitely many Maxwell--Boltzmann distributions with inverse temperaturesbounded below by a positive universal constant. Frankl's original conjecturecorresponds to zero inverse temperature.

The union-closed sets conjecture for non-uniform distributions