A Proof of Talagrand's Creating Large Sets Conjecture

Talagrand conjectured that if a family of sets $\mathcal{F}$ over $X=\{1,2,\cdots ,N\}$ is of large measure, then constant times of unions of sets in $\mathcal{F}$ will cover a large portion of the power set of $X$. This conjecture is a central open problem at the intersection of combinatorics and probability theory, and was described by Talagrand as a personal favorite. This paper provides a proof confirming this conjecture.