In Apostol's "Mathematical Analysis", Page 328 (see the image below and the underlined sentence),
why does the Lebesgue integral (41) exist for ?
The definition of convolution is as follows:
Then this is just because the product of two Riemann-integrable functions is Riemann-integrable (note that a Riemann-integrable function on [0,a] is bounded, contrary to a Lebesgue-integrable function on [0,a]). And if g is Riemann-integrable on [0,a], then so is for any , because it is obtained by symmetry from , which is R-integrable on (remember g is zero on ).