Let [ a, b] ⊂ R and let N be a sequence of positive linear operators from C n[a, b] ( nϵ Z +) to C[ a, b]. The convergence of L j to the identity operator I is closely related to the weak convergence of a sequence of finite measure μ j , to the unit (Dirac) measure δ x 0 , x 0 ϵ [ a, b]. New estimates are given for the remainder ¦∝ [a,b]ƒdμ j − ƒ(x 0)¦, where ƒ ϵ C n([a, b]) . Using moment methods, Shisha-Mond-type best or nearly best upper bounds are established for various choices of [ a, b], n and given moments of μ j . Some of them lead to attainable inequalities. The optimal functions/measures are typically spline functions and finitely supported measures. The corresponding inequalities involve the first modulus of continuity of ƒ (n) (the nth derivative of ƒ) or a modification of it. Several applications of these results are given.