Let $\mu$ be a probability measure on $[0,\infty)$ that is not degenerated ($\mu(0) < 1$) and $f$ be a bounded function on $[0,\infty)$. Show that pointwise $$f * \mu^{*n} \rightarrow 0$$ where $*$ denotes convolution $$(f * \mu)(t) = \int_0^t f(t-u) d \mu(u)$$ and $$\mu^{*n} = \mu * \dots * \mu\quad n\text{ times}$$
Edit: Added that $\mu$ is not degenerated. Edit2: Clarified that $f$ is defined on the positive reals.