  A non-negative function f, defined on the real line or on a half-line, is
said to be directly Riemann integrable (d.R.i.) if the upper and lower Riemann
sums 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 a
Lebesgue-integrable function f, very mild conditions are enough to ensure that
some n-fold convolution of f with itself is d.R.i.. Applications to renewal
theory and to local limit theorems are discussed.
