Almost almost Collatz

Let Col(x) = 3x+1 if x is odd, and x/2 if x is even. The infamous Collatz conjecture states that for every positive integer x, Col^n(x) = 1 for some n. Recently, Tao proved that for any function f(x) increasing to infinity, and almost all positive integers x (in the sense of logarithmic density), Col^n(x) < f(x) for some n. We give an overview of his beautiful proof, which almost manages to construct a nontrivial invariant measure for Col(x) (constructing one exactly is impossible unless the Collatz conjecture is false).