$\newcommand\vpi\varphi\newcommand\F{\mathcal F}$Suppose that $|g|\le C$ $\mu$-almost everywhere for some real $C>0$. Let $f_1:=\Re f\,1(\Re f>0)$, $f_2:=\Re f\,1(-\Re f>0)$, $f_3:=\Im f\,1(\Im f>0)$, $f_4:=\Im f\,1(-\Im f>0)$, so that the functions $f_1,\dots,f_4$ are $\ge0$ and $f=f_1-f_2+i(f_3-f_4)$.
It suffices to show that $\sup_{f\in\F}\int f_j |g|<\infty$ for each $j=1,2,3,4$. But $\vpi:=1(\Re f>0)$ is a simple function. So, $$\sup_{f\in\F}\int f_1 |g|\le C\sup_{f\in\F}\int f_1 =C\sup_{f\in\F}\int \Re f \vpi \\ =C\sup_{f\in\F}\Re\int f \vpi \le C\sup_{f\in\F}\Big|\int f \vpi\Big| <\infty, $$ so that $\sup_{f\in\F}\int f_1 |g|<\infty$. Similarly, $\sup_{f\in\F}\int f_j |g|<\infty$ for each $j=2,3,4$. $\quad\Box$