Approximate modularity: Kalton's constant is not smaller than 3

Kalton and Roberts [Trans. Amer. Math. Soc., 278 (1983), 803-816] proved that there exists a universal constant K ⩽ 44.5 such that for every set algebra F and every 1-additive function ƒ : F → R there exists a finitely-additive signed measure μ defined on F such that |ƒ(A) - μ(A)| ⩽ K for any A ε F. The only known lower bound for the optimal value of K was found by Pawlik [Colloq. Math., 54 (1987), 163-164], who proved that this constant is not smaller than 1.5; we improve this bound to 3 already on a non-negative 1-additive function.