Probability with Measure

1.8 Null sets

Sets of measure zero play an important role in measure theory, and in advanced probability. In fact, there is a special terminology for them, which we introduce in this section. In a general measure space \((S,\Sigma ,m)\) we say that a set \(E\in \Sigma \) is a null set if \(m(E)=0\). If we need to specify which measure is involved we might say e.g. \(m\)-null or Lebesgue null. We’ve already seen some examples of Lebesgue null sets in Section 1.5.

Lemma 1.8.1 For each \(n\in \N \) let \(E_n\) be a null set. Then \(\bigcup _{n=1}^\infty E_n\) is a null set.

Proof: By Lemma 1.7.2 we have \(m(\bigcup _{n=1}^\infty E_n)\leq \sum _{n=1}^\infty m(E_n)=0\). ∎

If a set \(E\in \Sigma \) is such that its complement is null (i.e. \(m(S\sc E)=0\)) then we say that \(E\) has full measure. You can prove an analogue of Lemma 1.8.1 for sets of full measure in Exercise 1.8.

We say that a property holds for almost all \(x\in S\) if the set of \(x\) for which the property holds has full measure. This is best understood by example: in Section 1.5 we deduce that the rational numbers \(\Q \) were a Lebesgue null subset of \(\R \). Therefore the set of irrational numbers has full measure. We can rephrase this statement as ‘almost all \(x\in \R \) are irrational numbers’. If we needed to be specific that we meant to use Lebesgue measure, we might say ‘Lebesgue almost all \(x\in \R \) are irrational’.

Remark 1.8.2 \((\star )\) Those of you taking courses in topics such as topology, metric spaces and functional analysis may wonder what relationships exist between sets of full measure and dense sets, when using the Borel \(\sigma \)-field on some metric or topological space as in Remark 3.3.4. The surprising answer is that in general a dense set might not have full measure, and a set of full measure might not be dense. The perspectives of measure theory (i.e. a set of full measure) and topology (i.e. a dense set) turn out to be quite different.