A measurable null-code order bounds the constructible null union

Relativize the definition-code recursion of [F1] to the structures $(L_{\alpha }[x],\in ,x\cap L_{\alpha }[x])$. It gives a coherent setlike well-order ${<}_{L[x]}$ whose levels are initial segments. Every real $d\in L[x]$ belongs to a countable level: inside $L[x]$, close $\omega \cup \{x,d\}$ under the canonically least Skolem witnesses of a sufficiently large level. Formula codes and finite tuples canonically enumerate this hull, so no ambient choice is used; collapsing it and inducting through the relativized definition operation gives some countable $L_{\beta }[x]$ containing $d$. Consequently the real predecessors of $d$ are countable and all occur in one such level.

A real $e$ can therefore certify that it enumerates exactly $\{c\in {\omega }^{\omega }:c{<}_{L[x]}d\}$: it codes a well-founded extensional relation on $\omega$, its collapse as a correct countable $L_{\beta }[x]$ containing $d$, the canonical order computed there, and the enumerated predecessor segment. Well-foundedness is ${\mathrm{\Pi }}_1^1$; extensionality, the staged definition recursion, countable satisfaction and the displayed enumeration check are arithmetic in the code. Thus the certificate predicate ${\mathrm{Pred}}_x(d,e)$ is ${\mathrm{\Pi }}_1^1(x)$, and $c\in L[x]$ has the analogous form $\mathrm{\exists }q{\mathrm{Lev}}_x(c,q)$ with ${\mathrm{Lev}}_x$ ${\mathrm{\Pi }}_1^1(x)$. Correctness follows by collapse and induction on the coded hierarchy; completeness of the predecessor list follows from the initial-segment property in step 1.1. This is the choice-free relativized certificate behind the standard ${\mathrm{\Sigma }}_2^1(x)$ facts in the cited sources.

Put $G$ equal to the union of the null $G_{\delta }$ sets having codes in $L[x]$. For $u\in G$, let $d(u)$ be the ${<}_{L[x]}$-least such code containing $u$, and let $\xi (u)$ be its position among the null codes. Disjointifying by least code gives null layers ${\tilde{G}}_{\xi }$ with $G={\bigcup }_{\xi }{\tilde{G}}_{\xi }$.

Define $A(x)=\{(u,v)\in G\times G:\xi (u)<\xi (v)\}$. Equivalently, $(u,v)\in A(x)$ iff there are reals $c,d,e,q$ such that ${\mathrm{Lev}}_x(c,q)$ and ${\mathrm{Pred}}_x(d,e)$ hold, $c$ codes a null $G_{\delta }$ containing $v$, $d$ codes one containing $u$ but not $v$, and no code enumerated by $e$ codes a null $G_{\delta }$ containing $v$. Indeed these conditions say $d{<}_{L[x]}d(v)$ while $d$ contains $u$; conversely take $d=d(u)$ and $c=d(v)$. In particular the formula is false off $G\times G$ and on the diagonal.

In the formula of step 3.1, the four real witnesses may be folded into one. The two certificate predicates are ${\mathrm{\Pi }}_1^1(x)$ by step 2.1, while recognition and interpretation of the explicit null-$G_{\delta }$ codes and the bounded checks through $e$ are arithmetic. A leading existential real followed by this ${\mathrm{\Pi }}_1^1$ matrix is ${\mathrm{\Sigma }}_2^1(x)$ in the convention of [F2].

For $v\in G$, the horizontal section $A(x{)}^v=\{u:\xi (u)<\xi (v)\}$ is the union of the null sets coded by the predecessor list for $d(v)$ from step 2.1, hence is null by Countable Choice. For $v\mathrm{\notin }G$ the section is empty. Thus every horizontal section is completed-measurable and null.

Assume $A(x)$ is measurable for the completed product coin measure. Tonelli applied to its indicator and step 4.2 makes $A(x)$ product-null. The completed Fubini theorem then supplies a completed-measurable null set $Z$ such that for every $u\mathrm{\notin }Z$ the vertical section $A(x{)}_u$ is measurable and null. No measure is assigned to exceptional vertical sections.

If $G\subseteq Z$, then $G$ is null. Otherwise choose $u\in G\setminus Z$. The lower section $A(x{)}^u$ is null by step 4.2, the middle layer ${\tilde{G}}_{\xi (u)}$ lies in one null $G_{\delta }$, and the upper section $A(x{)}_u$ is null by step 5.1. Since $G=A(x{)}^u\cup {\tilde{G}}_{\xi (u)}\cup A(x{)}_u$, the finite union is null. This dichotomy never presupposes measurability of $G$.

The steps above prove the complexity and the nullity conclusion, which is the Statement.