Every set of reals in the Solovay model is Lebesgue measurable

Use F2 to choose a bounded $N$ containing F1's sole real definition parameter; the finitely many ordinal parameters require no localization. F4 gives an $N$-coded Borel set $B$ agreeing with $A$ on every $N$-random real. In the ambient final extension enumerate the $N$-coded null Borel sets as $(C_n{)}_{n<\omega }$, as in F3's proof, and let $c$ be the real Borel code of their union $C$. Every nonrandom real lies in $C$, so $A\mathrm{△}B\subseteq C$. The code $c$ generally need not lie in $N$, but F1 says that $M$ and the final extension have the same reals; hence $c\in M$.

The code of $B$ is a real of $N$, hence a real of the final extension; F1's same-reals conclusion puts that code in $M$ without requiring the false class inclusion $N\subseteq M$. Step 1.1 likewise puts the code of $C$ in $M$. F5 makes $B$ Borel and $C$ null internally and supplies completeness of the null ideal, so every subset of $C$ is measurable; hence $A=B\mathrm{△}(A\mathrm{△}B)$ is measurable. This includes $A=\mathrm{\varnothing }$, $A=\mathbb{R}$, and zero exception $C=\mathrm{\varnothing }$.