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 a version of 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.

翻译：关于并集封闭集猜想（由Péter Frankl于1979年提出）指出：对于任何非空且有限并集封闭的有限集合族（该族不只包含空集），存在一个元素至少出现在该族一半的集合中。我们证明了Frankl猜想的一个版本适用于服从无穷多种分布中任意一种的集合族。作为推论，在Frankl猜想的交集封闭等价表述中，我们得出该猜想对服从无穷多种麦克斯韦-玻尔兹曼分布中任意一种的集合族成立，其中逆温度参数存在一个严格正的下界通用常数。Frankl原始猜想对应逆温度为零的情形。