Anderson's lemma

Appendix CAlmost invariant measures Anderson’s Lemma. Let f be a density with respect to Lebesgue measureon R s , such that:(i) f (x) = f (−x) for all x;(ii) the high density region { f ≥ t} is convex, for each t.For C a convex set symmetric about the origin, let C(θ) ={x + θ : x ∈ C}.Then ∫ {x ∈ C(θ)} f (x) dx is maximized at θ = 0. Corollary. Let ρ be a nonnegative function on R s such that {ρ ≤ t} is convexand symmetric about the origin, for each t. Then ∫ f (x − α)ρ(x − β) dx isminimized when α = β.Proof. By a change of variables, it is enough to consider the case whereα = 0. Let R t ={ρ ≤ t}. Then∫∫∫f (x)ρ(x − β) dx = f (x){0 ≤ t