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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