# Meagre sets and null sets

There are two competing notions for describing a subset of the real numbers as being ‘small’:

• a null set is a subset of the reals with Lebesgue measure zero;
• a meagre set is a countable union of nowhere-dense sets.

Both of these properties are downward-closed: an arbitrary subset of a null set is itself a null set, and an arbitrary subset of a meagre set is again a meagre set. Moreover, countable unions of meagre sets are meagre, and countable unions of null sets are null.

These two notions of a set being ‘small’ are also wholly incompatible.

In particular, there exist fat Cantor sets, nowhere-dense closed subsets of the unit interval which have positive Lebesgue measure arbitrarily close to 1. If you take a countable union of these sets (say, with Lebesgue measures of 1/2, 3/4, 7/8, 15/16, and so forth), the result is a meagre subset of [0, 1] with Lebesgue measure 1. If you take a countable union of translates of this set, each one occupying [n, n+1] for each integer n, the result is a meagre subset of the reals whose complement is a null set.

Stated more succinctly, there is a meagre set A (the one we’ve just constructed) and a null set B (its complement) such that A and B are disjoint and their union is the whole real line.

Moreover, A is an $F_{\sigma}$ set (countable union of closed sets) and B is a $G_{\delta}$ set (countable intersection of open sets). It’s possible to prove that every meagre set is a subset of a meagre $F_{\sigma}$ set, and likewise every null set is a subset of a null $G_{\delta}$ set. This turns out to be an ingredient of the proof of…

### The Erdős-Sierpiński duality theorem

From what has been mentioned so far, there seems to be some abstract ‘duality’ between null and meagre sets. Erdős and Sierpiński proved, conditional on the continuum hypothesis, a beautiful result that makes this precise:

There exists an involution (self-inverse bijection) f on the set of reals such that {f(x) : x in C} is null if and only if C is meagre, and {f(x) : x in D} is meagre if and only if D is null.

This involution f is highly discontinuous, being constructed using transfinite induction up to $2^{\aleph_0}$ (which, by assuming the continuum hypothesis, is also equal to the first uncountable cardinal $\aleph_1$). Shingo Saito describes the construction in detail.

This entry was posted in Uncategorized. Bookmark the permalink.