Tag Archives: Sigma algebra

Clarification on the proof of Carathéodory’s lemma

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 \lambda be an outer measure on the measurable space (S,\mathcal{G}). Then the λ-sets in \mathcal{G} form a σ-algebra \mathcal{L} on which \lambda is countably additive, so that (S,\mathcal{L},\lambda) 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 \mathcal{C} of subsets of a set S is called a λ-system on S if

  1. S\in\mathcal{C},
  2. L\in\mathcal{C}\Rightarrow L^{c}\in\mathcal{C} (it is closed under complements),
  3. (\forall n\in\mathbb{N})L_n\in\mathcal{C} with L_i\cap L_j=\emptyset for i\neq j it holds that \underset{n\in\mathbb{N}}{\cup}L_n\in\mathcal{C} (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 \emptyset\in\mathcal{C} instead of S\in\mathcal{C} due to closure under complementarity, i.e. due to the second condition of the definition.


Lemma

If a collection of subsets of a set S is a λ-system and a π-system on S, it is also a σ-algebra on S.

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 S is also a λ-system on S as it can be trivially seen from the involved definitions.


Proof of the lemma

Let \mathcal{C} be a collection of subsets S that is both a λ-system and a π-system on S. To show that \mathcal{C} is a σ-algebra on S, it suffices that it is closed under countable unions.

Let B_n\in\mathcal{C},n\in\mathbb{N}. The main idea is to express the collection \{B_n: n\in\mathbb{N}\} as a collection \{L_n: n\in\mathbb{N}\} of pairwise disjoint sets (L_i\cap L_j=\emptyset for i\neq j) so that \underset{n\in\mathbb{N}}{\cup}B_n=\underset{n\in\mathbb{N}}{\cup}L_n. Along these lines, define L_n:=B_n\setminus \left(\overset{n-1}{\underset{k=1}{\cup}} B_k\right).

Obviously, \underset{n\in\mathbb{N}}{\cup}L_n\subseteq \underset{n\in\mathbb{N}}{\cup}B_n. To prove the converse set inequality, let x\in\underset{n\in\mathbb{N}}{\cup}B_n and assume that (\forall n\in\mathbb{N}) x\not\in B_n\setminus \left(\overset{n-1}{\underset{k=1}{\cup}} B_k\right). In this case, for each n\in\mathbb{N}, either x\not\in B_n or x\in B_n\cap\left(\overset{n-1}{\underset{k=1}{\cup}} B_k\right). There is at least one n_{*}\in\mathbb{N} such that x\in B_{n_{*}}\cap\left(\overset{n_{*}-1}{\underset{k=1}{\cup}} B_k\right), otherwise (\forall n\in\mathbb{N}) x\not\in B_n leads to the contradiction x\not\in\underset{n\in\mathbb{N}}{\cup}B_n. Let n_o\in\mathbb{N} be the minimum natural for which x\in B_{n_{o}}\cap\left(\overset{n_{o}-1}{\underset{k=1}{\cup}} B_k\right). In turn, x\in \overset{n_{o}-1}{\underset{k=1}{\cup}} B_k\Rightarrow (\exists i<n_o)x\in B_i. Due to n_o being the smallest natural for which x\in B_{n_{o}}\cap\left(\overset{n_{o}-1}{\underset{k=1}{\cup}} B_k\right), it is deduced that x\not\in B_i\cap\left(\overset{i-1}{\underset{k=1}{\cup}} B_k\right), hence x\not\in\left(\overset{i-1}{\underset{k=1}{\cup}} B_k\right). Thus, (\exists i\in\mathbb{N}) x\in B_i\setminus \left(\overset{i-1}{\underset{k=1}{\cup}} B_k\right), which is a contradiction. Thereby, \underset{n\in\mathbb{N}}{\cup}B_n\subseteq \underset{n\in\mathbb{N}}{\cup}L_n, and this establishes the equality \underset{n\in\mathbb{N}}{\cup}L_n= \underset{n\in\mathbb{N}}{\cup}B_n.

Assume that there are i,j\in\mathbb{N} with i\neq j and L_i \cap L_j \neq \emptyset. Let x\in L_i \cap L_j. Without loss of generality assume that i<j. Then x\in B_i with i<j, while x\in L_j=B_j\setminus \left(\overset{j-1}{\underset{k=1}{\cup}} B_k\right), which means that (\forall k\in\mathbb{N}) with k<j it holds that x\not\in B_k, so a contradiction has been reached. Thereby, the sets L_n, n\in\mathbb{N}, are pairwise disjoint.

It has thus been shown that the collection L_n:=B_n\setminus \left(\overset{n-1}{\underset{k=1}{\cup}} B_k\right) consists of pairwise disjoints sets that satisfy \underset{n\in\mathbb{N}}{\cup}L_n=\underset{n\in\mathbb{N}}{\cup}B_n.

Notice that L_n:=B_n\setminus \left(\overset{n-1}{\underset{k=1}{\cup}} B_k\right)=B_n \overset{n-1}{\underset{k=1}{\cap}}B_k^c. Since \mathcal{C} is a λ-system, B_k^c\in\mathcal{C} for the various k. Moreover, \mathcal{C} is a π-system, hence the finite intersection L_n=B_n \overset{n-1}{\underset{k=1}{\cap}}B_k^c is also in \mathcal{C}. Since the collection L_n:=B_n\setminus \left(\overset{n-1}{\underset{k=1}{\cup}} B_k\right) is a disjoint union of elements L_n\in\mathcal{C} and \mathcal{C} is a λ-system, it follows that the union \underset{n\in\mathbb{N}}{\cup}L_n= \underset{n\in\mathbb{N}}{\cup}B_n is also in \mathcal{C}.

Since the countable (but not necessarily disjoint) union \underset{n\in\mathbb{N}}{\cup}B_n of any collection \{B_n: n\in\mathbb{N}\} of sets B_n\in\mathcal{C} is also in \mathcal{C}, it follows that \mathcal{C} is a σ-algebra.


First clarification

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 \{L_n: n\in\mathbb{N}\} of disjoint sets L_n\in\mathcal{L} it holds that \underset{n\in\mathbb{N}}{\cup}L_n\in\mathcal{L}. The conclusion then extends to any such countable union of sets, disjoint or not.


Second clarification

It is mentioned in p. 197 of Williams’ book that from

\lambda (G)\ge\overset{n}{\underset{k=1}{\sum}}\lambda(L_k\cap G)+\lambda(L^c\cap G)

follows

\lambda (G)\ge\overset{\infty}{\underset{k=1}{\sum}}\lambda(L_k\cap G)+\lambda(L^c\cap G).

To see why this is the case, recall that the outer measure \lambda:\mathcal{G}\rightarrow [ 0, \infty ] takes in values in \lambda (G)\in[0, \infty] for any G\in\mathcal{G}.

Distinguish two cases. If \lambda (G)\in [0,\infty ) (i.e. if \lambda (G) is finite), then the sequence a_n=\overset{n}{\underset{k=1}{\sum}}\lambda(L_k\cap G),n\in\mathbb{N}, is a bounded increasing sequence, therefore it converges, which means that the limit \overset{\infty}{\underset{k=1}{\sum}}\lambda(L_k\cap G)=\underset{n\rightarrow\infty}{\lim}a_n < \infty exists, so taking limits leads from the former to the latter inequality in the book.

If \lambda (G)=\infty, then \infty \ge\overset{\infty}{\underset{k=1}{\sum}}\lambda(L_k\cap G)+\lambda(L^c\cap G) holds trivially.


Third clarification

To show that \lambda (L)=\underset{k\in\mathbb{N}}{\sum}\lambda (L_k) for L=\underset{k\in\mathbb{N}}{\cup}L_k, notice first that

\lambda (L)\le\underset{k\in\mathbb{N}}{\sum}\lambda (L_k)

follows from the countable subadditivity of the outer measure \lambda.

Moreover, setting G=L in equation (d) of p. 197 gives

\lambda (L)\ge\underset{k\in\mathbb{N}}{\sum}\lambda (L_k\cap L)+\lambda (L^c\cap L)=\underset{k\in\mathbb{N}}{\sum}\lambda (L_k),

which concludes the argument.

Clarification on algebras and σ-algebras of a set

David Williams makes a terminological remark on algebras in p. 16 of his book “Probability with martingales”; he mentions that an algebra \Sigma on a set S (defined ordinarily as a collection of subsets of S that contains S and that is closed under set complementation and closed under finite set unions) is “a true algebra in the algebraists’ sense”.

This remark emphasises two aspects of the concept of algebra \Sigma on a set S. Firstly, it means that \Sigma is an algebra over a field K. The field K contains exactly two elements, so it can be defined to be the subset K=\{0,1\} of integers. According to the definition of algebra over a field, \Sigma is a vector space equipped with a bilinear product. So any algebra (and consequently any σ-algebra) is a type of vector space of sets (subsets of S).

Secondly, consider the defining operations of an algebra \Sigma on a set S. The vector space addition is defined to be the symmetric difference A\Delta B:= (A\cup B)\setminus (A\cap B). The bilinear product, which turns the vector space to an algebra over \{0,1\}, is the set intersection A\cap B. It is straightforward to check that the product A\cap B satisfies right and left distributivity and compatibility with scalars. The two operations (addition and bilinear product) defined from the product space \Sigma\times\Sigma to the vector space \Sigma are both symmetric.

In summary, any algebra (and any σ-algebra) is a vector space of sets equipped with two symmetric set operations.