A note on directly Riemann integrable functions

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.

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