Solovay measure on all ground-set subsets in a supplied generic extension

Let $T=({\mathbb{B}}^I{)}^M$, a set in $M$. For $a\in T$ form ${\sigma }_a=\{⟨\check{i},a_i⟩:i\in I\}$. This is a name in $M$ by internal Replacement. F5 and valuation give $A(a):={\mathrm{val}}_G({\sigma }_a)=\{i\in I:a_i\in G\}$. Conversely, for any $A\in M[G]$ with $A\subseteq I$, take one name $\tau \in M$ whose valuation is $A$. Internal definability of the fixed atomic Boolean value gives the vector $a_i=\mathrm{\parallel }\check{i}\in \tau {\mathrm{\parallel }}^M$ in $T$. F3 says $a_i\in G$ iff $i\in A$, so $A=A(a)$. No simultaneous choice of a name for all such $A$ was used. The name $\dot{\mathcal{D}}=\{⟨{\sigma }_a,1⟩:a\in T\}\in M$ evaluates exactly to $\mathcal{D}$, so this full collection belongs to $M[G]$ without an appeal to its Power Set axiom.

Internally apply F1 to all $a\in T$, and select measurable $[0,1]$-valued representatives $h_a$ by F8. Their assignment is a set function in $M$. F2 supplies the associated rational-cut names ${\rho }_a$ as a set-indexed assignment. For any two names $s,t$, the name $P(s,t)=\{⟨s,1⟩,⟨t,1⟩\}$ evaluates to the unordered pair of their valuations since $1\in G$. Therefore $K(s,t)=P(P(s,s),P(s,t))$ evaluates to their Kuratowski ordered pair. These finite constructions are internal set operations and yield names in $M$. Define the graph name $\dot{\eta }=\{⟨K({\sigma }_a,{\rho }_a),1⟩:a\in T\}$. Its valuation is the relation $\{(A(a),(h_a{)}_G):a\in T\}$, which belongs to $M[G]$.

If $A(a)=A(b)$, then for every $i\in I$ either both $a_i,b_i$ belong to $G$ or neither does. Ultrafilterhood puts $e_i=(a_i\wedge b_i)\vee (\mathrm{\neg }{a}_i\wedge \mathrm{\neg }{b}_i)$ in $G$. The family $(e_i{)}_{i\in I}$ is a ground family, so its meet $c$ belongs to $G$ by F4, even when $I$ is large. Since $c\le e_i$, Boolean distributivity gives $c\wedge a_i=c\wedge b_i$ for every $i$. F1 locality gives $h_a=h_b$ almost everywhere on $c$, and F2 gives $(h_a{)}_G=(h_b{)}_G$. Consequently the relation from step 1.2 is a function on exactly $\mathcal{D}$. Null modifications of the selected representatives do not change its values, by F2. Every value is between zero and one, by the same evaluation lemma.

Let $\beta \in M$ be an ordinal and let $f=(A_{\xi }{)}_{\xi <\beta }\in M[G]$ be a function with values in $\mathcal{D}$. Take a single name $\tau \in M$ for its graph. For $(i,\xi )\in I\times \beta$, define $a_i^{\xi }=\mathrm{\parallel }\mathrm{\exists }y\mathrm{\exists }p(p=⟨\check{\xi },y⟩\wedge p\in \tau \wedge \check{i}\in y){\mathrm{\parallel }}^M$, where the ordered-pair expression abbreviates its membership-language definition. This is a ground table by internal Replacement and fixed-formula definability in F3. F6 ensures transitivity of $M[G]$, F5 supplies $i,\xi$ there, and step 1.2 supplies its finite-pair closure. Thus F7 identifies the displayed pair formula with actual ordered pairs. Since the valuation of $\tau$ is the actual graph $f$, F3 proves $a_i^{\xi }\in G$ iff $i\in A_{\xi }$. Hence all family members are represented simultaneously by this single ground table. This conclusion does not assume that the family $f$ itself belongs to $M$.

For a ground $Z\subseteq I$, use the vector $a_i=1$ on $Z$ and zero elsewhere, which evaluates to $Z$. F1 says its density is almost everywhere constant one or zero according as $Z\in U$ or not. F2 evaluates those constants to themselves, proving the extension assertion. A proper ultrafilter contains $I$ and excludes the empty set, so in particular $\eta (I)=1$ and $\eta (\mathrm{\varnothing })=0$. Properness also rules out the degenerate case $I=\mathrm{\varnothing }$.

Put $a_i={\bigvee }_{\xi <\beta }{a}_i^{\xi }$ internally. F4 gives $a_i\in G$ iff some $a_i^{\xi }\in G$, because each coordinate's joined family belongs to $M$. Thus $A(a)={\bigcup }_{\xi <\beta }{A}_{\xi }$, and this union lies in $\mathcal{D}$ by step 1.1. For $\beta =0$ the vector is constantly zero and the union empty. This works for any ground ordinal $\beta$; no completeness property of $U$ has been used in this union calculation.

Suppose now $\beta =\omega$ and the $A_n$ are disjoint. For each $i\in I$ and $n<r<\omega$, the element $\mathrm{\neg }(a_i^n\wedge a_i^r)$ lies in $G$, since otherwise ultrafilterhood would put both coefficients in $G$ and hence $i$ in both sets. These elements form one ground family, so F4 puts their common meet $c$ in $G$. On this $c$, every coordinatewise intersection $c\wedge a_i^n\wedge a_i^r$ is zero. F1 then proves $h_a={\sum }_n{h}_{a^n}$ almost everywhere on $c$, where $a$ is the union vector of step 3.2. The density representatives are a ground sequence of bounded nonnegative functions, so F2 gives $(h_a{)}_G={\sum }_n(h_{a^n}{)}_G$. Step 2.1 identifies these values with $\eta ({\bigcup }_n{A}_n)$ and $\eta (A_n)$, respectively. This proves the asserted countable additivity for every extension sequence in the statement.

Finally let $\beta <\kappa$ and suppose every $\eta (A_{\xi })=0$. For the ground table of step 2.2, F2 says each zero-set class $z(a^{\xi })$ belongs to $G$. The density assignment and this table are in $M$, so this is a ground family of zero-set classes. F4 places $c={\bigwedge }_{\xi <\beta }z(a^{\xi })$ in $G$. Internally F1 gives $c\le z(a)$ for the coordinatewise union vector, since $\beta <\kappa$ there. Upward closure and the reverse zero-test direction in F2 give $\eta (A(a))=(h_a{)}_G=0$. Step 3.2 identifies $A(a)$ with the required union. For the empty family F1 uses its empty-meet inequality, and step 3.1 already gives the same conclusion; the singleton case gives the original null set. This proves the full stated indexed null closure without taking an uncountable union of exceptional measurable null sets.

The names in steps 1.1–1.2 witness that both the full subset collection and its measure graph are elements of $M[G]$. Steps 2.2–4.2 cover new indexed families by one ground Boolean table, rather than by an assumption that the new family is ground. Their only cardinal comparison is the ground comparison $\beta <\kappa$; preservation of $\kappa$ and its relation to the new continuum are not conclusions here. AC was used for the ground set of density representatives and the prerequisites in F1, as specified in F8. Every other selected name was one existential witness. The argument proves exactly the supplied-model statement, with no inference from it to formal Con.