Corollary

Let $g$ be a $\mu$-measurable function on X. The multiplication operator $m_g$ on $L^2(X, \mu)$ is self-adjoint (resp. positive ) if and only if the function $g$ is locally $\mu$-almost everywhere real-valued (resp. locally $\mu$-almost everywhere positive).

The first assertion results from the proposition. If $g$ is locally $\mu$-almost everywhere positive, one has $\langle f|m_g(f)\rangle =\int_Xg|f|^2d\mu\geqslant 0$ for every $f\in L^2(X, \mu)$, hence the partial operator $m_g$ is positive.

Conversely, if $m_g$ is positive, then its spectrum is contained in $\mathbf{R}_+$ (prop. 17 of IV, p. 248); since this is the $\mu$-essential image of $g$ (prop. 22, a)), this means that $g$ is locally $\mu$-almost everywhere positive.