Integrable Functions and the Integral with Respect to a Finite Signed Measure

In the setting of Radon Measures on Locally Compact Metric Spaces: Standing Notation, let $(X,\mathcal{F})$ be a measurable space and let $\nu$ be a finite signed measure on $(X,\mathcal{F})$, with total variation $|\nu|$, a finite measure by Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability (i), and with Jordan decomposition $\nu^{+},\nu^{-}$, formed with a Hahn set for $\nu$.

(i) Integrable functions. A function $f:X\to\mathbb{R}$ is integrable with respect to $\nu$ if $f\in\mathcal{L}^{1}(|\nu|)$. The set of these functions is $\mathcal{L}^{1}(\nu)=\mathcal{L}^{1}(|\nu|)$.

(ii) The integral. For $f\in\mathcal{L}^{1}(\nu)$, which lies in $\mathcal{L}^{1}(\nu^{+})$ and in $\mathcal{L}^{1}(\nu^{-})$ by Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability (viii), the integral of $f$ with respect to $\nu$ is the real number

$$\int_{X}f\,d\nu=\int_{X}f\,d\nu^{+}-\int_{X}f\,d\nu^{-},$$

also written $\int f\,d\nu$ (by Hahn and Jordan Decompositions of a Finite Signed Measure (ii) it does not depend on the Hahn set).