Taken from Conway's A course in functional analysis Chapter 3 Section 7 Problem 2
Problem Statement: Let $X$ be a set and $\Omega$ a $\sigma$-algebra of subsets of $X$. Suppose $\mu$ is a complex-valued countably additive measure defined on $\Omega$ such that $\|\mu\| = \mu(X) < \infty$. Show that $\mu(\Delta) \geq 0$ for every $\Delta$ in $\Omega$
Given the section this problem is in, I assume we are supposed to use Banach limits to solve this problem. However, Banach limits are linear functionals acting on $l^\infty$ spaces which makes it hard for me to see how to apply it to this problem.
According to a consequence of Radon-Nikodym Theorem (see Rudin, Real & Complex Analysis, Theorem 6.12), there exists a measurable function $h:X\to \mathbb C$, with $|h(x)|=1$, for all $x\in X$, such that $$ d\mu=h\,d|\mu|, $$ and hence $$ \int_X d|\mu|=\|\mu\|=\mu(X)=\int_X d\mu=\int_X d\mu=\int_X h\, d|\mu| $$ Hence $$ 0=\int_X (1-h)\, d|\mu|=\int_X (1-\mathrm{Re}\,h)\, d|\mu|, $$ and hence Re$\,h(x)=1$, almost everywhere, which implies that Im$\,h(x)=0$, almost everywhere, since $|h|=1$. Thus $h(x)=1$, almost everywhere, and hence $\mu$ is a positive measure.