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Source. E. Glazer, Erdős Problem 501 after adding $\omega_2$ random reals, draft rev10, the opening paragraph of Section 4 and Lemma 4.1 (Borel reading), physical pp. 4--5, in the eight-page PDF held by its library source card, Glazer (2026). The source gives a six-line proof; the version here expands it and names the standard facts it rests on.

Standing. This is an author-recorded reconstruction. It is not an independent review and changes no status and assigns no tier. The facts labeled (R1)--(R5) below are imported, not proved here.

Conventions and imported facts

For a set $\Theta$ of coordinates, let $\mu_\Theta$ be the completion of the product of the fair-coin measures on $2^\Theta$. The measure algebra $\mathbb B(\Theta)$ is the Boolean algebra of $\mu_\Theta$-measurable subsets of $2^\Theta$ modulo $\mu_\Theta$-null sets; a condition is a nonzero element, and forcing with $\mathbb B(\Theta)$ over a ground model $M$ is forcing with this complete Boolean algebra, with Boolean values $\|\varphi\|\in\mathbb B(\Theta)$. The source writes $\mathbb B_{\omega_2}$ for $\mathbb B(\omega_2\times\omega)$. For $S\subseteq\Theta$ and $u\in2^\Theta$, $u\restriction S\in2^S$ is the restriction.

The following standard facts are used on this page and the later forcing pages. For random forcing and measure algebras the source lists K. Kunen, Random and Cohen reals, Handbook of Set-Theoretic Topology (1984), 887--911; the Boolean-valued forcing facts are in T. Jech, Set Theory, third millennium edition, Chapters 14--15, and the descriptive set theory in A. S. Kechris, Classical Descriptive Set Theory (1995).

• (R1) Countable supports. $\mathbb B(\Theta)$ is a complete Boolean algebra with the countable chain condition, and every $\mu_\Theta$-measurable set is $\mu_\Theta$-almost equal to a set of the form $W\times2^{\Theta\setminus S}$ with $S\subseteq\Theta$ countable and $W\subseteq2^S$ Borel. A countable $S$ supports an element of $\mathbb B(\Theta)$ if the element has such a representative, and supports a name if it supports every Boolean value occurring in the name (the source's wording); a support may always be enlarged.
• (R2) The generic point. $\dot G$ denotes the canonical name for the point $u_G\in2^\Theta$ with $u_G(\theta)=1$ if and only if $[\{u:u(\theta)=1\}]\in G$. For countable $S\subseteq\Theta$ and Borel $W\subseteq2^S$ coded in $M$, $\|\dot G\restriction S\in W\|=[W\times2^{\Theta\setminus S}]$, where $W$ is reinterpreted in the extension from its code. In particular a Borel $W\subseteq2^S$ is $\mu_S$-null if and only if $\Vdash\dot G\restriction S\notin W$, and the condition $[W\times2^{\Theta\setminus S}]$, when nonzero, forces $\dot G\restriction S\in W$.
• (R3) Forcing theorem and maximum principle. If a condition $q$ forces $\exists x\,\varphi(x)$, there is a name $\dot x$ with $q\Vdash\varphi(\dot x)$; names may be mixed along a partition of unity.
• (R4) Absoluteness. Standard Borel spaces, Borel sets and Borel maps coded in $M$ are reinterpreted in $M[G]$ from the same codes, a coded preimage, complement or countable union being reinterpreted as the preimage, complement or union of the reinterpretations; Borel statements about points of $M$ are absolute between $M$ and $M[G]$; and a $\Pi^1_1$ statement about the coded objects that holds in $M$ holds in $M[G]$ (Mostowski's absoluteness theorem, T. Jech, Set Theory, third millennium edition, Chapter 25), in particular that a coded Borel map is injective, carries a coded set into a coded set, or is inverse to another coded map. A name for an element of a standard Borel space $X$ is a name $\dot z$ with $\Vdash\dot z\in X$ for the reinterpreted $X$.
• (R5) Borel isomorphism. Every standard Borel space is Borel isomorphic to a Borel subset of $2^\omega$.

Statement

The following is provable in ZFC. Let $X$ be a standard Borel space and $\dot z$ a $\mathbb B(\Theta)$-name for an element of $X$. There are a countable $S\subseteq\Theta$ and a Borel map $F\colon2^S\to X$ such that

$$\Vdash_{\mathbb B(\Theta)}\dot z=F(\dot G\restriction S).$$

We say that such an $S$ reads $\dot z$ through $F$. If $S$ reads $\dot z$ through $F$ and $S\subseteq S'$ is countable, then $S'$ reads $\dot z$ through $u\mapsto F(u\restriction S)$, because $(\dot G\restriction S')\restriction S=\dot G\restriction S$.

Proof

The case $X=2^\omega$. For each $n<\omega$ the Boolean value $b_n=\|\dot z(n)=1\|$ lies in $\mathbb B(\Theta)$. By (R1) choose a countable $S_n\subseteq\Theta$ and a Borel $W_n\subseteq2^{S_n}$ with $b_n=[W_n\times2^{\Theta\setminus S_n}]$. Put $S=\bigcup_nS_n$, countable, and define $F\colon2^S\to2^\omega$ by

$$F(u)(n)=1\iff u\restriction S_n\in W_n.$$

Each coordinate of $F$ is the indicator of a Borel set, so $F$ is Borel. By (R2), $\|\dot G\restriction S_n\in W_n\|=b_n=\|\dot z(n)=1\|$ for every $n$, so $\Vdash\dot z(n)=F(\dot G\restriction S)(n)$ for every $n$, and hence $\Vdash\dot z=F(\dot G\restriction S)$.

General $X$. By (R5) fix a Borel isomorphism $\iota$ of $X$ onto a Borel set $X'\subseteq2^\omega$. Then $\iota(\dot z)$ is a name for an element of $2^\omega$; by the first case obtain countable $S$ and Borel $F_0\colon2^S\to2^\omega$ with $\Vdash\iota(\dot z)=F_0(\dot G\restriction S)$. The set $N=F_0^{-1}(2^\omega\setminus X')$ is Borel, and by (R2) and (R4) $\|\dot G\restriction S\in N\|=\|\iota(\dot z)\notin X'\|=0$, so $N$ is $\mu_S$-null. Fix $x_0\in X$ and define $F(u)=\iota^{-1}(F_0(u))$ for $u\notin N$ and $F(u)=x_0$ for $u\in N$. Then $F$ is Borel, and since $\Vdash\dot G\restriction S\notin N$, $\Vdash\dot z=F(\dot G\restriction S)$.

Boundary. The lemma is applied in Proposition 4.4 to names for elements of $\mathcal O^{\mathbb Z}$ and in Lemma 4.5 to a name for a Borel code.