Applications of a comeager null set

There are a number of classic examples¹ of simple sets of reals that are :comeager and :null. These examples are usually used to illustrate the orthogonality of measure and category: a set can be large in the measure-theoretic sense but small in Baire-category (i.e., more topological) terms, and vice versa. However I’ve recently come into a few applications of these sets beyond just intuition-checking.

Size of ground model reals

Fact. After adding a Cohen real, the set of ground model reals is null. Dually, after adding a random real, the set of ground model reals is meager.

Proof. The real line is provably the union of a meager set \(A\) and a null set \(B\), and this partition is absolute across forcing extensions (once we re-interpret the Borel codes). If \(c\) is Cohen, then for every ground model real \(r\), we have \(r\in c-B\) (:why?). Since measure is preserved by translation, then \(c-B\) is null, so the ground model reals are a subset of a null set, hence null. The dual argument works for random reals.

Size of generic reals

Fact. After adding a Cohen real, the set of Cohen reals over the ground model is null. Dually, after adding a random real, the set of random reals over the ground model is meager.

Proof. Similar reasoning as above. Every Cohen real will avoid the meager set \(A\), so the set of Cohen reals is a subset of the null set \(B\). Dually for random reals.

Size of Luzin and Sierpinski sets when both exist

Theorem. (Rothberger 1938) If there is a :Luzin set and a :Sierpinski set, then both have cardinality \(\omega_1\). Consequently, the continuum hypothesis holds. (See also Theorem 2.3 in Miller’s chapter Special Subsets of the Real Linein the Handbook of Set-Theoric Topoplogy.)

The comeager null set is used in the following lemma:

Lemma. If \(X\) is not meager and has cardinality \(\kappa\), then the real line is the union of \(\kappa\) many null sets (similarly with null and meager swapped).

Proof of Lemma. Let \(A\) be a comeager null set. The real line can be covered by sets of the form \(\{x+A \mid x\in X\}\): for if some \(z\) is not in any of these sets, then the comeager set \(z-A\) is disjoint from \(X\), contradicting the non-meagerness of \(X\).

Proof of Theorem. Since the subset of a Luzin (resp. Sierpinski) set is still Luzin (resp. Sierpinski), then there must be such sets of size \(\omega_1\). Suppose \(L\) is such a Luzin set. By the lemma, since \(L\) is non-meager, the real line is the union of \(\|L\|=\omega_1\) many null sets \(N_\alpha\). It follows that any Sierpinski set must have size at most \(\omega_1\), this is because if \(S\) is Sierpinski, then it intersects each of those null sets in countably many points, and since \(S=\mathbb{R}\cap S = (\bigcup_{\alpha<\omega_1}N_\alpha) \cap S=\bigcup_{\alpha<\omega_1}(N_\alpha \cap S)\), we have \(\|S\|\leq \omega_1\cdot \omega = \omega_1\). Similarly for Luzin sets. Thus both have size exactly \(\omega_1\), and so the continuum hypothesis holds.

:x comeager

A set is comeager iff its complement is meager (a.k.a. of first category). Meager means it can be written as the countable union of nowhere dense sets.

:x null

This is an other way of saying Lebesgue measure zero.

:x why1

\(c\) avoids every ground model-coded meager set, by Solovay’s characterization of Cohen-genericity. Thus \(c\notin r+A\), this means \(c\in r+B\), and hence \(r\in c-B\).

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