Newelski 1987 infinite free set small measure set mappings

../

---

Ludomir Newelski, Janusz Pawlikowski and Witold Seredyński, Infinite free set for small measure set mappings, Proc. Amer. Math. Soc. 100 (1987), no. 2, 335--339; DOI 10.1090/S0002-9939-1987-0884475-3. Received by the editors January 6, 1986.

The copy read for this card is the publisher's scan of the five printed pages with an OCR text layer (physical PDF p. $n$ is printed p. $334+n$); the formulas in the text layer are garbled, so the Theorem was checked on the page image of p. 336. Provenance: obtained in a survey download of September 2026; the download URL was not recorded; 468,585 bytes. The scan prints "©1987 American Mathematical Society" in a footnote on its first page (printed p. 335), every other right reserved.

Read status. Claims checked: the Lemma, the Theorem and Corollaries (1)--(4) were read clause by clause (the Theorem on the page image, the rest in the text layer); the proofs were read but not verified.

Contents

Corollary (1) is the statement that concerns the second question of Problem 501.

• Definitions (p. 335): "A set $A\subset X$ is free for a function $F\colon X\to\mathcal P(X)$ if $x\notin F(y)$ for any distinct $x,y\in A$." In Problem 38(B) of [EH], Erdős and Hajnal asked what size of free set is forced when $F$ sends each real number to a closed set of measure less than $1$; Gładysz [G] had proved the existence of a free pair under an integral condition.
• Lemma (p. 335; proof p. 336): for a $\sigma$-finite measure space $(X,\mathcal X,\mu)$, a second countable space $Y$ with a $\sigma$-finite measure $\nu$ on the $\sigma$-field generated by its closed sets, and $A\subseteq X\times Y$ with all sections $A_x$ closed, $\overline{\int}_Y\mu_*(A^y)\,d\nu(y)\le\overline{\int}_X\nu(A_x)\,d\mu(x)$ (upper integrals on both sides, with the inner measure $\mu_*$ on the left).
• Theorem (p. 336; proof pp. 336--338): let $X$ be second countable, $\mathcal X$ the $\sigma$-field generated by its closed sets, $\mu$ a $\sigma$-finite measure on $\mathcal X$, and $F\subseteq X\times X$ with every section $F_x$ closed. (a) If $\mu(X)<\infty$, $2\le k<\omega$, and for every $x\in X$ and $B\subseteq X$ with $\mu^*(B)>0$

$$\mu(X)>(k-1)\Bigl(\mu^*(\{x\})+\frac2{\mu^*(B)}\overline{\int}_X \chi_B(t)\,\mu^*(F_t\cap B)\,d\mu(t)\Bigr), \qquad (*)$$

then there is a $k$-element free set; for $k=2$ it suffices to check $(*)$ for $B=X$. (b) There is an infinite free set provided (i) $\mu(X)=\infty$ and $\mu(F_x)<C$ for all $x$, for some $C<\infty$, or (ii) $\mu(F_x\cup\{x\})=0$ for all $x$.

• Corollary (1) (p. 338): "On the real line consider a map $x\to F(x)$, where $F(x)$ is closed of measure less than 1. Then there is an infinite free set." The paper adds: "This answers Problem 38(B) of Erdös and Hajnal [EH]."
• Corollary (2) (p. 338): every set mapping on the unit interval whose values $F(x)$ are closed null sets has an infinite free set (Erdős [E] had proved this for nowhere dense values).
• Corollary (3) (p. 338): if $\mu$ vanishes on points and $2k-2<\mu(X)<\infty$, $k\ge2$, then for $F(x)$ closed of measure less than $1$ there is a $k$-element free set, indeed disjoint sets $B_1,\dots,B_k$ of positive outer measure any selector of which is free.
• Corollary (4) (p. 338): for $F\colon\{0,\dots,n-1\}\to\mathcal P(\{0,\dots,n-1\})$ with $|F(x)|\le m$, a $k$-element free set exists if and only if $k<1+n/(2m+1)$; the Notes (p. 339) say this shows that the Theorem's assumptions are in some sense essential.
• Added in proof (p. 339): D. H. Fremlin's weakening of $(*)$ for atomless $\mu$, and a forthcoming paper of the second author, Pawlikowski, "Half Fubini theorem", which discusses a strengthened version of the Lemma.

Compiled scope

The whole five-page paper was read, with the Theorem checked on the page image and the rest in the text layer; the proofs were followed but not verified. Nothing here is independently reviewed.

Bears on. #501, whose second question, whether closed $A_x$ of measure less than $1$ force an independent set of size $3$, is answered by Corollary (1), which gives an infinite free set on the real line without any boundedness assumption and which the paper presents as the answer to Problem 38(B) of Erdős and Hajnal; the first question, on bounded sets of outer measure less than $1$ that need not be closed, is outside the Theorem's closed-sections hypothesis.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.