Suppose that (X, p) is a sermonized space, {x_k}ⁿ_k=0 is a linearly independent system of elements in X, {c _k}ⁿ_k=0 is a sequence of linear bounded functionals such that c _k (x_l) = δ_kl, R_n,r (x) = n ∑ k=0 (1 - (k/n+1)^r) c _k(x)x_k are the Riesz sums. We prove general assertions concerning estimates from above for the values of semiadditive functionals Φ : X → ℝ₊ by deviations of the Riesz sums p(x - R _n,r (x)). Bibliography: 6 titles.