Almost all sets of nonnegative integers and their small perturbations are not sumsets

Fix α ∈ (0, 1/3). We show that, from a topological point of view, almost all sets A ⊆ N have the property that, if A 0 = A for all but o(n α ) elements, then A 0 is not a nontrivial sumset B + C. In particular, almost all A are totally irreducible. In addition, we prove that the measure analogue holds with α = 1.