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Source. E. Glazer, Erdős Problem 501 after adding $\omega_2$ random reals, draft rev10, Lemma 4.2 (factorization, stated on p. 5 and cited to the literature) and Lemma 4.5 (fresh-profile fullness), physical p. 6, in the eight-page PDF held by its library source card, Glazer (2026).

Standing. This is an author-recorded reconstruction. It is not an independent review and changes no status and assigns no tier. Lemma 4.2 is imported below exactly as the source states it; the conventions (R1)--(R5) of the Lemma 4.1 page are assumed. One measurability point that the source folds into "by Fubini" is labeled and closed with a repository-supplied remark.

Imported: Lemma 4.2 (factorization)

The following is provable in ZFC. If $\Sigma$ and $\Gamma$ are disjoint coordinate sets, then $\mathbb B(\Sigma\cup\Gamma)$ is the completed product of $\mathbb B(\Sigma)$ and $\mathbb B(\Gamma)$: $\mu_{\Sigma\cup\Gamma}$ is the completion of the product measure $\mu_\Sigma\times\mu_\Gamma$ on $2^\Sigma\times2^\Gamma=2^{\Sigma\cup\Gamma}$, so Fubini and Tonelli apply to $\mu_{\Sigma\cup\Gamma}$-measurable sets. In particular, if $G$ is $\mathbb B(\Sigma\cup\Gamma)$-generic over $M$, then $G\restriction\Sigma$ is $\mathbb B(\Sigma)$-generic over $M$ and $G\restriction\Gamma$ is $\mathbb B(\Gamma)$-generic over $M[G\restriction\Sigma]$; the analogous statement holds for finite and countable families of pairwise disjoint coordinate sets. The source cites M. Laczkovich and A. W. Miller, Measurability of functions with approximately continuous vertical sections and measurable horizontal sections, Colloq. Math. 69 (1996), 299--308, Fact 1 in the proof of Lemma 8. That paper is not held and its proof is not reconstructed here.

Definitions

Let $(P_\alpha)_{\alpha\in J}$ be an uncountable family of pairwise disjoint countable coordinate sets, each identified with a fixed countable set $P$ through a bijection $\pi_\alpha\colon P\to P_\alpha$, and let $\Gamma$ be a further coordinate set disjoint from every petal. Put $\Theta'=\bigcup_{\alpha\in J}P_\alpha\cup\Gamma$ and force with $\mathbb B(\Theta')$. The normalized generic point on $P_\alpha$ is the name

$$\dot z_\alpha=(\dot G\restriction P_\alpha)\circ\pi_\alpha\in2^P.$$

In the extension, $\nu$ is the fair-coin product measure on $2^P$ and $\nu^*$ its outer measure; $\dot Z$ names $\{\dot z_\alpha:\alpha\in J\}$.

Statement

The following is provable in ZFC. With the notation above,

$$\mathbb B(\Theta')\Vdash\nu^*\bigl(\{\dot z_\alpha:\alpha\in J\}\bigr)=1$$

(the source's (4.3)).

Proof

Reduction. In any model, a set $Z\subseteq2^P$ has $\nu^*(Z)<1$ if and only if some Borel $B\subseteq2^P$ has $\nu(B)>0$ and $B\cap Z=\varnothing$ (take $B$ to be the complement of a Borel superset of $Z$ of measure below one, and conversely). Suppose the statement fails. Then some condition $q_0$ forces that such a $B$ exists for $\dot Z$, and, by inner regularity of $\nu$ in the extension, that some closed such $B$ exists. Fix an enumeration $(U_n)_{n<\omega}$ of the basic clopen subsets of $2^P$ and, for $c\in2^\omega$, put $K_c=2^P\setminus\bigcup\{U_n:c(n)=1\}$: every $c\in2^\omega$ codes a closed set, every closed $K\subseteq2^P$ is $K_c$ for $c=\{n:U_n\cap K=\varnothing\}$, and the relation $v\in K_c$ is closed in $(c,v)$. By the maximum principle (R3) there is a name $\dot c$ for an element of $2^\omega$, mixed with a fixed default off $q_0$ so that $\Vdash\dot c\in2^\omega$; write $\dot B=K_{\dot c}$, so that

$$q_0\Vdash\nu(\dot B)>0\ \text{ and }\ \dot B\cap\dot Z=\varnothing.$$

Since $q_0$ forces that some positive rational lies below $\nu(\dot B)$, strengthen $q_0$ to a condition $q$ deciding one: fix a rational $\varepsilon>0$ with $q\Vdash\nu(\dot B)>\varepsilon$ (the source's (4.4); the source keeps $\dot B$ Borel, given by a Borel code, see the labeled point below).

A fresh petal. By Lemma 4.1 and (R1), a countable $T\subseteq\Theta'$ supports $q$ and reads $\dot c$ through a Borel map $F\colon2^T\to2^\omega$. The petals are pairwise disjoint, so only countably many of them meet the countable set $T$; as $J$ is uncountable, choose $\alpha\in J$ with $P_\alpha\cap T=\varnothing$.

Factor over $T$ and $P_\alpha$. For $t\in2^T$ let $B_t=K_{F(t)}\subseteq2^P$, a closed set, and consider

$$W=\{(t,v)\in2^T\times2^P:v\in B_t\},\qquad W'=\{u\in2^{\Theta'}:u\restriction T\in q,\ (u\restriction T,(u\restriction P_\alpha)\circ\pi_\alpha)\in W\},$$

where $q$ is identified with a Borel subset of $2^T$ supporting it. The set $W$ is Borel, being $\bigcap_n\{(t,v):F(t)(n)=0\text{ or }v\notin U_n\}$, and so is the base of the cylinder $W'$. By (R2), (R4) and the identity $\dot z_\alpha=(\dot G\restriction P_\alpha)\circ\pi_\alpha$, the Boolean value $\|\dot z_\alpha\in\dot B\|\wedge q$ is the class $[W']$. By Lemma 4.2 applied to $T$, $P_\alpha$ and the rest of $\Theta'$, together with Tonelli's theorem for the completed product,

$$\mu_{\Theta'}(W')=\int_q\nu(B_t)\,d\mu_T(t)$$

(the source's (4.5), with $\nu(B_t)$ computed through $\pi_\alpha$ on $2^{P_\alpha}$). For $\mu_T$-almost every $t\in q$, $\nu(B_t)>\varepsilon$: otherwise the Borel set $\{t\in q:\nu(B_t)\le\varepsilon\}$, Borel because $t\mapsto\nu(K_{F(t)})=1-\sup_N\nu\bigl(\bigcup\{U_n:n<N,\ F(t)(n)=1\}\bigr)$ is a Borel function, has positive measure, and as a condition of $\mathbb B(T)\subseteq\mathbb B(\Theta')$ below $q$ it forces, by (R2), $\nu(\dot B)\le\varepsilon$, contradicting the choice of $q$. Hence

$$\mu_{\Theta'}(W')\ge\varepsilon\,\mu_T(q)>0.$$

Contradiction. So $r:=[W']$ is a nonzero condition, $r\le q$, and $r\Vdash\dot z_\alpha\in\dot B$. But $q\Vdash\dot B\cap\dot Z=\varnothing$ and $r\le q$, so $r\Vdash\dot z_\alpha\notin\dot B$. This is impossible. Therefore no condition forces $\nu^*(\dot Z)<1$; equivalently, the complement of $\dot Z$ contains no positive Borel set, and $\Vdash\nu^*(\dot Z)=1$.

Labeled point (compilation remark). The source takes $\dot B$ to be a Borel set given by a Borel code and folds both the measurability of $W$ and the transfer of the ground-model measure computation into the forcing relation into "by Fubini". The proof above shrinks $\dot B$ to a closed set first, a step the source does not take and not an author-issued correction: with closed codes every $c\in2^\omega$ is a code, $W$ is Borel, both appeals to (R2) are within its hypotheses, and (R1)--(R4), Lemma 4.2 and Tonelli's theorem for the completed product suffice. With general Borel codes the set of codes is coanalytic and not Borel, so $W$, $W'$ and $\{t\in q:\nu(B_t)\le\varepsilon\}$ are only coanalytic; they are still universally measurable (A. S. Kechris, Classical Descriptive Set Theory (1995), Chapters 29 and 35), so the display for $\mu_{\Theta'}(W')$ stands, but the two appeals to (R2) then need an import beyond (R1)--(R5): for a coanalytic $C\subseteq2^S$ coded in $M$, take in $M$ a Borel $C_0\subseteq C$ with $\mu_S(C\setminus C_0)=0$; the $\Pi^1_1$ inclusion $C_0\subseteq C$ holds in $M[G]$ by Mostowski's absoluteness theorem (T. Jech, Set Theory, third millennium edition, Chapter 25), so the condition $[C_0]=[C]$ forces $\dot G\restriction S\in C$ by (R2), which is what both steps use.

Boundary. The lemma is applied in Theorem 5.1 to the petals of Proposition 4.4, with $\Gamma$ the remaining coordinates of $\omega_2\times\omega$; there the uncountability of $J$ is $|J|=\omega_2$ in the ground model, which is all the proof uses.