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Source. E. Glazer, Erdős Problem 501 after adding $\omega_2$ random reals, draft rev10, Theorem 5.1 (forcing interface), physical pp. 6--7, in the eight-page PDF held by its library source card, Glazer (2026). It uses Definition 3.1 from the Theorem 3.2 page, Proposition 4.4 and Lemma 4.5.

Standing. This is an author-recorded reconstruction. It is not an independent review and changes no status and assigns no tier. The conventions (R1)--(R5) of the Lemma 4.1 page are assumed; outer regularity of Lebesgue outer measure is used as a definition. The instance of the map $\rho$ under Definitions is a compilation fill.

Definitions

$\kappa=\omega_2$, $\Theta=\kappa\times\omega$, $D_\alpha=\{\alpha\}\times\omega$ and $\mathbb B=\mathbb B(\Theta)$, as on the Proposition 4.4 page; $\mathcal O$, $U(c)$, $I_m=[m,m+1)$ and profile certificates as on the Theorem 3.2 page. Fix a Borel measure-preserving map $\rho\colon2^\omega\to[0,1)$ (fair-coin measure to Lebesgue measure) with null point fibers. One instance: $\rho(r)=\sum_nr(n)2^{-n-1}$, redefined as $0$ on the null set of sequences that are eventually $1$; it pushes the fair-coin measure to Lebesgue measure on $[0,1)$, and each fiber is countable, hence null. Through the enumeration $n\mapsto(\alpha,n)$ of $D_\alpha$, $2^{D_\alpha}$ is identified with $2^\omega$, and through the enumeration $\langle d_n\rangle$ of $D$ from Proposition 4.4, so is $2^D$.

A name $\dot{\mathcal A}=(\dot A_y)_{y\in\mathbb R}$ is a name for a function from the reals of the extension to subsets of those reals; for a name $\dot x$ for a real, $\dot A_{\dot x}$ names its value at $\dot x$.

Statement

$$\mathrm{ZFC}+\mathrm{CH}\vdash\ \mathbb B_{\omega_2}\Vdash \forall\mathcal A\,\bigl[(\forall y\in\mathbb R\ \lambda^*(A_y)<1) \longrightarrow\mathrm{Prof}(\mathcal A)\bigr]$$

(the source's (5.1)).

Proof

Work in a ground model $M\models\mathrm{ZFC}+\mathrm{CH}$ and let $p\in\mathbb B$ force

$$\dot{\mathcal A}=(\dot A_y)_{y\in\mathbb R}\ \text{ satisfies }\ \lambda^*(\dot A_y)<1\ \text{ for every }y\in\mathbb R$$

(the source's (5.2)). We show that $p$ forces $\mathrm{Prof}(\dot{\mathcal A})$.

Random points. For $\alpha<\kappa$ let $\dot r_\alpha=\dot G\restriction D_\alpha\in2^\omega$, the random real read from the block $D_\alpha$, and for $m\in\mathbb Z$ put

$$\dot x_{\alpha,m}=m+\rho(\dot r_\alpha)\in I_m$$

(the source's (5.3)).

Names for envelopes. Fix $\alpha<\kappa$ and $m\in\mathbb Z$. In any extension by a generic containing $p$, $\lambda^*(A_{x_{\alpha,m}})<1$, so by the definition of outer measure there is an open set of measure below one containing $A_{x_{\alpha,m}}$, and every open set has a code. By the maximum principle (R3) there is a name $\dot c_{\alpha,m}$ with

$$p\Vdash\dot A_{\dot x_{\alpha,m}}\subseteq U(\dot c_{\alpha,m}) \ \text{ and }\ \lambda(U(\dot c_{\alpha,m}))<1$$

(the source's (5.4)). Mix $\dot c_{\alpha,m}$ with a fixed default code below the complement of $p$, so that the top condition forces $\dot c_{\alpha,m}\in\mathcal O$; this makes it a name for an element of the standard Borel space $\mathcal O$ in the sense of (R4). Bundle the countable sequence into

$$\dot w_\alpha=\langle\dot c_{\alpha,m}:m\in\mathbb Z\rangle \in\mathcal O^{\mathbb Z},$$

a name for an element of the standard Borel space $\mathcal O^{\mathbb Z}$.

Homogeneous reading. Apply Proposition 4.4 (in $M$, which satisfies CH) to $p$, $X=\mathcal O^{\mathbb Z}$ and $(\dot w_\alpha)_{\alpha<\kappa}$. Obtain $J\subseteq\kappa$ of size $\kappa$, the root $R$, the petals $(P_\alpha)_{\alpha\in J}$ with $D_\alpha\subseteq P_\alpha$, the pair $D\subseteq P$ with bijections $\pi_\alpha\colon P\to P_\alpha$ sending $d_n$ to $(\alpha,n)$, and a Borel map

$$F=\langle F_m:m\in\mathbb Z\rangle\colon2^R\times2^P\to\mathcal O^{\mathbb Z}$$

(the source's (5.5)) with $\Vdash\dot w_\alpha=F(\dot G\restriction R,(\dot G\restriction P_\alpha)\circ\pi_\alpha)$ for $\alpha\in J$. The set $P$ is countably infinite, since $D\subseteq P$ is in bijection with $D_\alpha$.

The certificate. Let $G\ni p$ be $\mathbb B$-generic over $M$ and work in $M[G]$. Set $g=G\restriction R\in2^R$ and

$$z_\alpha=(G\restriction P_\alpha)\circ\pi_\alpha\in\Omega:=2^P \qquad(\alpha\in J)$$

(the source's (5.6)). Let $\nu$ be the fair-coin product measure on $\Omega$, a standard Borel probability space, and put $Z=\{z_\alpha:\alpha\in J\}$. Lemma 4.5, applied in $M$ to the petals $(P_\alpha)_{\alpha\in J}$ (uncountable in $M$) and $\Gamma=\Theta\setminus\bigcup_{\alpha\in J}P_\alpha$, gives $\nu^*(Z)=1$ in $M[G]$ (the source's (5.7)). This is (P1).

For $z\in\Omega$ and $m\in\mathbb Z$ define the raw code $c^0_m(z)=F_m(g,z)$, a Borel function of $z$ (a section of the Borel map $F_m$ at the fixed $g$), and the truncated code

$$c_m(z)=\begin{cases} c^0_m(z),&\lambda(U(c^0_m(z)))<1,\\ c_\varnothing,&\lambda(U(c^0_m(z)))\ge1 \end{cases}$$

(the source's (5.8)), where $U(c_\varnothing)=\varnothing$. Since $c\mapsto\lambda(U(c))$ is Borel, the case split is over a Borel set and $c_m\colon\Omega\to\mathcal O$ is Borel; and $\lambda(U(c_m(z)))<1$ for every $z\in\Omega$ (the source's (5.9)). This is (P3). The truncation is what makes (P3) hold on all of $\Omega$ rather than only on $Z$.

Define

$$x_m(z)=m+\rho(z\restriction D)$$

(the source's (5.10)). It is Borel with values in $I_m$. The restriction $z\mapsto z\restriction D$ pushes $\nu$ to the fair-coin measure on $2^D$, and $\rho$ pushes that to Lebesgue measure on $[0,1)$, so for Borel $B\subseteq\mathbb R$, $\nu(x_m^{-1}(B))=\lambda((B-m)\cap[0,1))=\lambda(B\cap I_m)$. This is (P2).

Finally let $z=z_\alpha\in Z$ with $\alpha\in J$. In $M[G]$, since $p\in G$ and $\dot w_\alpha$ is read by $F$,

$$\dot c_{\alpha,m}^G=F_m(g,z_\alpha)=c^0_m(z_\alpha)\qquad(m\in\mathbb Z).$$

Moreover $x_m(z_\alpha)=\dot x_{\alpha,m}^G$: by the choice of $\pi_\alpha$, $(z_\alpha\restriction D)(d_n)=G(\alpha,n)=\dot r_\alpha^G(n)$ for every $n$, so $z_\alpha\restriction D$ is $\dot r_\alpha^G$ under the fixed identifications, and $x_m(z_\alpha)=m+\rho(\dot r_\alpha^G)$. Hence, by the forced statement (5.4) with $p\in G$,

$$\lambda(U(c^0_m(z_\alpha)))<1\quad\text{and}\quad A_{x_m(z_\alpha)}\subseteq U(c^0_m(z_\alpha)).$$

The first inequality says that the truncation leaves the code at $z_\alpha$ unchanged, $c_m(z_\alpha)=c^0_m(z_\alpha)$, so

$$A_{x_m(z_\alpha)}\subseteq U(c_m(z_\alpha))$$

(the source's (5.11)). This is (P4).

Thus $(\Omega,\nu)$, $Z$, $\langle x_m,c_m:m\in\mathbb Z\rangle$ is a profile certificate for $\mathcal A=\dot{\mathcal A}^G$ in $M[G]$.

Closing the quantifiers. Every generic $G\ni p$ satisfies $\mathrm{Prof}(\dot{\mathcal A}^G)$, so $p\Vdash\mathrm{Prof}(\dot{\mathcal A})$ by the forcing theorem. If some condition forced $(\forall y\ \lambda^*(\dot A_y)<1)\wedge\neg\mathrm{Prof}(\dot{\mathcal A})$, the argument applied to that condition would contradict it; hence the top condition forces the implication for $\dot{\mathcal A}$. Every family in $M[G]$ has a name, so the universal statement (5.1) follows.

Boundary. CH is used only through Proposition 4.4. Lemma 4.5 and Theorem 3.2 are ZFC theorems. The assembly into Theorem 1.1 is on the Theorem 1.1 page.