My professor used the following lemma in the proof that $L^1(X,\mu)^* = L^\infty(X,\mu)$ but left the proof as an exercise.
Lemma. Assume that $(X,\mathcal A, \mu)$ is a measure space and $g \in L^1(X,\mu)$. Suppose there is a constant $C > 0$ such that $$\left| \int_A g \, d\mu \,\right| \le C \mu(A)$$ for all $A \in \mathcal A$. Then $g \in L^\infty(X,\mu)$ and $\lVert g\rVert_\infty \le C$.
If $g$ is real-valued, the proof is relatively straight-forward but I am struggling to see how to prove this when $g$ is complex-valued. In that case, I can prove $\lVert g\rVert_\infty \le 2C$ but I am wondering how to get $\lVert g\rVert_\infty \le C$.