We have:
1. $\{f_n\} \subset L^1(E, \Sigma, \mu)$
2. $g_n \subset L^\infty(E, \Sigma, \mu)$
3. $\|f_n-f\|_{L^1}\to0$
4. $g_n \to g \text{ }\mu\text{-a.e.}$
5. $\{g_n\}$ is uniformly bounded.
Then show that $\|f_n g_n- f g\|_{L^1} \to 0$
I've been trying to apply the usual trick of
$$|f_n g_n-fg|=|f_n g_n-g_nf+g_nf-fg|\leq|g_n||f_n-f|+|f||g_n-g|$$
and then integrating. Under the assumptions all the terms (and factors) except $|g_n-g|$ either go to zero in the norm (or the factors are bounded). The problem with $|g_n-g|$ is that to make the $L^1$ norm go to zero I need to assume that $\mu(E)<\infty$ to apply the Dominated Convergence Theorem, adding one more assumption. Is there any other way without assuming this.