Suppose that $g_n(x)$ is an increasing sequence of summable functions and $g_n(x)\to1$ as $n\to\infty$. Given $f\geq0$ integrable the result follows$$\lim_{n\to\infty}\int_\Omega g_n(x)f(x)\,dx=\int_\Omega f(x)\,dx.$$
It is easy to show that such result holds using the Monotone Convergence Theorem, but what I am wondering, is if whether the $f\geq0$ integrable part is essential. Intuitively, $f$ would need to be integrable, otherwise the right hand of the equation would not exist, but why must it be that $f\ge0$. Is there a proof of this?
If we separate $f$ into $f=f_+-f_-$ with $f_\pm\ge0$, your generalization follows under the caveat that at most one of $\int f_\pm\,\mathrm dx$ is $+\infty$.