If $g \in L^1(\mathbb{R})$ and $(f_n)$ is a sequence of measurable functions converging to $f$ a.e where $|f_n| \leq 1$, then $g * f_n \to g * f$ uniformly on every compact set where $g*f = \int_{\mathbb{R}} g(x-y)f(y)dy$.
Using Egoroff's theorem, the sequence converges almost uniformly on every compact set; however, I am having difficulty extending this to the entire set.
edit:
$f_n \to f$ almost uniformly on a set $D$ if, for every $\epsilon > 0$, there is a set $E$ such that $\mu(E) < \epsilon$ and $f_n \to f$ uniformly on $D \setminus E$.
It's clear from dominated convergence that $f_n*g\to f*g$ pointwise.
Recall that $g\in L^1$ implies that $$\lim_{h\to0}\int|g(t)-g(t+h)|\,dt=0.$$
It follows that the sequence $(f_n*g)$ is equicontinuous. And pointwise convergence plus equicontinuity implies uniform convergence (at least on compact sets).