a reference for a well-known theorem of analysis on convolution of a bounded and an $\ell$-integrable function is wanted

Question

On page 12 of Nummelin's General Markov chains and Non-Negative Operators some 'well-known theorem of analysis* is mentiones, "according to which the convolution of a bounded and of an $\ell$-integrable function on $(\mathbb R, \mathscr R)$ is continuous". Can someone give a reference where I can find this theorem to check the rigorous statement? I doubt that a bounded function is always measurable and hence the product of a bounded and of an integrable function is neccesserely integrable...