Conditional expectation exists by radon nikodym

Let $\mu =P{\mathrm{\mid }}_{\mathcal{G}}$ and ${\nu }_{\pm }(A)={\int }_A{X}^{\pm }dP$. Restricting the measures in [F2] from $\mathcal{F}$ to $\mathcal{G}$ makes ${\nu }_{\pm }$ finite positive measures, with total mass at most $E\mathrm{\mid }X\mathrm{\mid }$. If $\mu (A)=0$, [F7] gives ${\nu }_{\pm }(A)=0$, so ${\nu }_{\pm }\ll \mu$. The constant exhaustion $\mathrm{\Omega }$ has finite $\mu$ and finite variation for both positive measures.

Apply [F3] separately to $(\mu ,{\nu }_{+})$ and $(\mu ,{\nu }_{-})$. Its AC hypothesis is [F6]; its exhaustion and absolute continuity were checked in step 1.1. It supplies real, $\mathcal{G}$-measurable integrable $f_{+},f_{-}$ with ${\int }_A{f}_{\pm }d\mu ={\nu }_{\pm }(A)$. They are nonnegative almost everywhere: on $N_{\pm }=\{f_{\pm }<0\}$ positivity of ${\nu }_{\pm }$ and nonpositivity of the integral force ${\int }_{N_{\pm }}(-f_{\pm })=0$, so [F5] makes $N_{\pm }$ null. Set them to zero there. These are $\mathcal{G}$-measurable null sets, so the modification is legitimate without completing $\mathcal{G}$. The current RN interface already supplies real integrable densities, so no infinite density is subtracted.

Put $Y=f_{+}-f_{-}$. It is real, $\mathcal{G}$-measurable and integrable, and for every $A\in \mathcal{G}$, ${\int }_{A}YdP={\nu }_{+}(A)-{\nu }_{-}(A)={\int }_A(X^{+}-X^{-})dP={\int }_{A}XdP$, with only finite subtractions. Thus [F1] makes $Y$ the required version.