Consider a lebesgue measure λ on [0 , 1] and let 0 ≤ h ∈ L 1 (λ). We construct a measure µ such that kµk ≤ 2. khkL1(λ) , µ⊥λ and µ (E) = 0, for every Carleson set E in [0 , 1] and converges weak-star to h dλ.