Characterization of the convergence of weighted averages of sequences and functions

The weighted averages of a sequence (ck), ck ∈ ℂ, with respect to the weights (pk), pk ≥ 0, with {fx135-1} are defined by {fx135-2} while the weighted average of a measurable function f: ℝ+ → ℂ with respect to the weight function p(t) ≥ 0 with {fx135-3}. Under mild assumptions on the weights, we give necessary and sufficient conditions under which the finite limit σn → L as n → ∞ or σ(t) → L as t → ∞ exists, respectively. These characterizations may find applications in probability theory.