Carathéodory’s lemma, as it appears in p. 197 (appendix A1) of the book “Probability with martingales” by David Williams, is stated as follows:
Let be an outer measure on the measurable space . Then the λ-sets in form a σ-algebra on which is countably additive, so that is a measure space.
Three aspects of the proof of Carathéodory’s lemma provided in Williams’ book are clarified in this blog post.
Definition of λ-system
The concept of λ-system, which is used implicitly but it is not defined in Williams’ book, is introduced in this post.
A collection of subsets of a set is called a λ-system on if
- (it is closed under complements),
- with for it holds that (it is closed under countable disjoint unions).
Not that the only difference between a λ-system and a σ-algebra is that the former is closed under countable disjoint unions while the latter is closed under countable unions. Moreover, the first condition on the definition of a λ-system could be alternatively set to instead of due to closure under complementarity, i.e. due to the second condition of the definition.
If a collection of subsets of a set is a λ-system and a π-system on , it is also a σ-algebra on .
This lemma is used without being proved in Williams’ book for proving Carathéodory’s lemma. In what follows, the lemma will be proved before proceeding with the proof of Carathéodory’s lemma.
Although not relevant to subsequent developments, it is mentioned that a σ-algebra on a set is also a λ-system on as it can be trivially seen from the involved definitions.
Proof of the lemma
Let be a collection of subsets that is both a λ-system and a π-system on . To show that is a σ-algebra on , it suffices that it is closed under countable unions.
Let . The main idea is to express the collection as a collection of pairwise disjoint sets ( for ) so that . Along these lines, define .
Obviously, . To prove the converse set inequality, let and assume that . In this case, for each , either or . There is at least one such that , otherwise leads to the contradiction . Let be the minimum natural for which . In turn, . Due to being the smallest natural for which , it is deduced that , hence . Thus, , which is a contradiction. Thereby, , and this establishes the equality .
Assume that there are with and . Let . Without loss of generality assume that . Then with , while , which means that with it holds that , so a contradiction has been reached. Thereby, the sets , are pairwise disjoint.
It has thus been shown that the collection consists of pairwise disjoints sets that satisfy .
Notice that . Since is a λ-system, for the various . Moreover, is a π-system, hence the finite intersection is also in . Since the collection is a disjoint union of elements and is a λ-system, it follows that the union is also in .
Since the countable (but not necessarily disjoint) union of any collection of sets is also in , it follows that is a σ-algebra.
The above lemma explains why the proof of Carathéodory’s lemma in Williams’ book states that it suffices to show that for a countable collection of disjoint sets it holds that . The conclusion then extends to any such countable union of sets, disjoint or not.
It is mentioned in p. 197 of Williams’ book that from
To see why this is the case, recall that the outer measure takes in values in for any .
Distinguish two cases. If (i.e. if is finite), then the sequence , is a bounded increasing sequence, therefore it converges, which means that the limit exists, so taking limits leads from the former to the latter inequality in the book.
If , then holds trivially.
To show that for , notice first that
follows from the countable subadditivity of the outer measure .
Moreover, setting in equation (d) of p. 197 gives
which concludes the argument.