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Source. E. Glazer, Erdős Problem 501 after adding $\omega_2$ random reals, draft rev10, Lemma 2.2 (preservation step), physical p. 3, 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.

Definitions

The setting and the sections $E_t$, $E^s$, the column bound $\mu(E^s)\le K$ and the set $Q(C)$ are those of Lemma 2.1.

Statement

The following is provable in ZFC. In addition to the hypotheses of Lemma 2.1, let $x\colon S\to\mathbb R$ be $\Sigma$-measurable with every fiber null:

$$\mu(\{t\in S:x(t)=a\})=0\qquad(a\in\mathbb R)$$

(the source's (2.4)). If $C\subseteq S$ is measurable with $\mu(C)=\infty$ and $t\in Q(C)$, then

$$C'=C\setminus\bigl(E_t\cup E^t\cup\{s\in S:x(s)=x(t)\}\bigr)$$

(the source's (2.5)) is measurable and $\mu(C')=\infty$.

Proof

The sections $E_t$ and $E^t$ of the measurable set $E$ lie in $\Sigma$, and the fiber $\{s:x(s)=x(t)\}=x^{-1}(\{x(t)\})$ lies in $\Sigma$ because $x$ is measurable. So $C'\in\Sigma$.

Write $C'=(C\setminus E_t)\setminus N$ with $N=E^t\cup\{s:x(s)=x(t)\}$. Since $t\in Q(C)$, $\mu(C\setminus E_t)=\infty$. By the column bound, $\mu(E^t)\le K<\infty$, and the fiber is null by hypothesis, so $\mu(N)\le K$. If $\mu(C')$ were finite, then

$$\mu(C\setminus E_t)\le\mu(C')+\mu(N)<\infty,$$

a contradiction. Hence $\mu(C')=\infty$.

Boundary. The lemma is the inductive step of the recursion in Theorem 3.2; the fiber removal there is what makes the selected reals pairwise distinct.