The union-closed sets conjecture for non-uniform distributions

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 not consisting of only the empty set, there is an element that is in at least half of the sets in the family. We prove Frankl's conjecture for families distributed according to any one of infinitely many distributions. As a corollary, in the intersection-closed reformulation of Frankl's conjecture, we obtain that it is true for families distributed according to any one of infinitely many Maxwell--Boltzmann distributions with inverse temperatures bounded below by a positive universal constant. Frankl's original conjecture corresponds to zero inverse temperature.