A note on directly Riemann integrable functions

摘要

A non-negative function f, defined on the real line or on a half-line, issaid to be directly Riemann integrable (d.R.i.) if the upper and lower Riemannsums of f over the whole (unbounded) domain converge to the same finite limit,as the mesh of the partition vanishes. In this note we show that, for aLebesgue-integrable function f, very mild conditions are enough to ensure thatsome n-fold convolution of f with itself is d.R.i.. Applications to renewaltheory and to local limit theorems are discussed.