I'm reading this question about signed measure. Assume that $\lambda$ is a finite signed measure and have a Jordan decomposition $\lambda = \lambda^+ - \lambda^-$. Then $$ \int f \mathrm d \lambda := \int f \mathrm d \lambda^+ - \int f \mathrm d \lambda^-, $$ if both integrals are finite.
However, I could not find a textbook that define integration with respect to a sign measure. Could you please suggest me some textbook references?