Deriving Dynkin’s theorem from the monotone class theorem


One of the first major results of any measure theory course is that the Lebesgue measure is the unique measure that assigns to each interval its length. A related result is the extension theorem1

Every probability measure on a field F\mcF can be extended to a unique probability measure on the generated σ\sigma-field.

Here, a probability measure PP on a field means it has σ\sigma-additivity: if A1,A2,FA_1,A_2,\ldots\in\mcF and nAnF\bigcup_n A_n\in\mcF, then P(nAn)=nP(An)P(\bigcup_n A_n) = \sum_n P(A_n). We can prove the preceding result as follows: the intervals of [0,1][0, 1] generate a field of disjoint unions of intervals. We can define a natural measure on this interval that is finitely additive, which can be shown to be, in fact, countably additive, so it has a unique extension to the generated σ\sigma-field of Borel sets B\mcB. This unique extension is the Lebesgue measure on [0,1][0,1].

We can construct this extension by defining the outer measure, designating subsets of Ω\Omega as measurable based on Carathéodory’s criterion, and then showing that all sets in the generated σ\sigma-field are measurable.

Vector art of a Shiba Inu dog wearing black sunglasses

Cool! What about uniqueness?

π\pi and λ\lambda-systems

Let Ω\Omega be a set. A π\pi-system is a class of subsets of Ω\Omega closed under finite intersections. A λ\lambda-system is a class that contains Ω\Omega, is closed under complement and under disjoint countable unions. These two definitions constitute the essence of a σ\sigma-field: a class is a σ\sigma-field if and only if it is both a π\pi-system and a λ\lambda-system. Of special interest here is Dynkin’s π-λ\pi\text{-}\lambda theorem:

If a π\pi-system P\msP is a subset of a λ\lambda-system L\msL, then σ(P)L\sigma(\msP) \subset\msL.

How does this help us? Suppose P1P_1 and P2P_2 are probability measures on σ(F)\sigma(\mcF) that agree on F\mcF. Let L\msL be the maximal subset of σ(F)\sigma(\mcF) on which these two measures agree; L\msL can be shown to be a λ\lambda-system. Since the field F\mcF is certainly a π\pi-system and FL\mcF\subset\msL, Dynkin’s theorem gives us σ(F)L\sigma(\msF)\subset\msL, so the two measures agree on the generated σ\sigma-field.

Monotone classes

A class M\msM of subsets of Ω\Omega is monotone if it is closed under monotone unions and intersections:

Similar to π\pi-systems and λ\lambda-systems, the requirements of a field and a monotone class capture the essence of a σ\sigma-field: a class is a σ\sigma-field if and only if it is both a field and a monotone class. Perhaps propitiously, we have an analogous version of Dynkin’s theorem called Halmos’s monotone class theorem:

If a field F\mcF is a subset of a monotone class M\msM, then σ(F)M\sigma(\mcF)\subset\msM.

Can this be used to prove the uniqueness in the extension theorem? Yes, since all λ\lambda-systems are monotone. The class L\msL above is a monotone class containing a field F\mcF, so it contains σ(F)\sigma(\mcF) too.

Vector art of a Shiba Inu dog wearing black sunglasses

But how do we prove either of these theorems? Given their similarity, are they equivalent to each other?

They are indeed equivalent, and it is a fun exercise to prove this. First, we need some groundworks on the representation of a generated field.

Generated field

For a non-empty class A\msA of subsets of Ω\Omega, the field generated by A\msA, or f(A)f(\msA), can be described explicitly: its members have the form i=1mj=1niAij\bigcup_{i=1}^m \bigcap_{j=1}^{n_i} A_{ij}, where AijAA_{ij}\in\msA or AijcAA_{ij}^c\in\msA for each ii and jj, and where the mm sets j=1ni\bigcap_{j=1}^{n_i} are disjoint.

Proof. If a set AΩA\subset\Omega satisfies AAA\in\msA or AcAA^c\in\msA, we call it A\msA-compatible (just a made-up name). Let F(A)F(\msA) be the class of sets of the given form, i.e., a finite union of finite intersections of A\msA-compatible sets. We can see that A\msA is a subset of F(A)F(\msA). Suppose S=i=1mj=1niAijS = \bigcup_{i=1}^m\bigcap_{j=1}^{n_i} A_{ij} is a member of F(A)F(\msA). Its complement is2

Sc=(i=1mj=1niAij)c=i=1mj=1niAijc=jMi=1mAijic,S^c = \left(\bigcup_{i=1}^m\bigcap_{j=1}^{n_i}A_{ij}\right)^c = \bigcap_{i=1}^m\bigcup_{j=1}^{n_i}A_{ij}^c = \bigcup_{\bm{j}\in M}\bigcap_{i=1}^m A_{i\bm{j}_i}^c,

where M={j=(j1,,jm) ⁣:ji[1,ni]}M=\{\bm{j}=(j_1,\ldots,j_m)\colon j_i\in [1, n_i]\}. Note that AijicA_{ij_i}^c is the complement of some AijA_{ij}, so it is A\msA-compatible. Hence, F(A)F(\msA) contains ScS^c and is closed under complementation.

Now, consider the intersection of two members of F(A)F(\msA):

(i=1mj=1niAij)(k=1pl=1qkBkl)=1im1kp(j=1niAijl=1qkBkl)=1im1kpCiDk.\left(\bigcup_{i=1}^m\bigcap_{j=1}^{n_i} A_{ij}\right) \cap \left(\bigcup_{k=1}^p\bigcap_{l=1}^{q_k} B_{kl}\right) = \bigcup_{\substack{1\le i\le m\\ 1\le k\le p}} \left(\bigcap_{j=1}^{n_i}A_{ij}\cap \bigcap_{l=1}^{q_k}B_{kl}\right) = \bigcup_{\substack{1\le i\le m\\ 1\le k\le p}} C_i\cap D_k.

Each CiDkC_i\cap D_k is a finite intersection of A\msA-compatible sets, and if (i,k)(i,k)(i,k)\ne (i',k'), then CiDkC_i\cap D_k is disjoint from CiDkC_{i'}\cap D_{k'} (since at least one index must be different, say, iii\ne i', then CiC_i and CiC_{i'} are disjoint). Thus, the above set is also a member of F(A)F(\msA), which means F(A)F(\msA) is closed under finite intersection and is a field. Since it also contains A\msA, we have f(A)F(A)f(\msA)\subset F(\msA).

For any field F\msF containing A\msA, every A\msA-compatible set is a member of F\msF, and so the finite union of finite intersections of A\msA-compatible sets is also a member of F\msF. In other words, F(A)F(\msA) is a subset of any field containing A\msA, so it is a subset of f(A)f(\msA). Thus, f(A)=F(A)f(\msA)=F(\msA). \blacksquare

Equivalence

We now come to the main task: deriving Dynkin’s π\pi-λ\lambda theorem from the monotone class theorem. Suppose the λ\lambda-system L\msL contains the π\pi-system P\msP. We want to show that L\msL contains the σ\sigma-field generated by P\msP, but the monotone class theorem only gives us that if L\msL already contains a field. Hence, we shall fill in this gap by proving that L\msL contains the field generated by P\msP.

If the λ\lambda-system L\msL contains the π\pi-system P\msP, then L\msL contains f(P)f(\msP).

Proof. By the previous section, f(P)f(\msP) consists of sets of the form i=1mj=1niAij\bigcup_{i=1}^m\bigcap_{j=1}^{n_i} A_{ij}, where each AijA_{ij} is P\msP-compatible and the mm sets j=1niAij\bigcap_{j=1}^{n_i}A_{ij} are disjoint. Since L\msL is closed under countable disjoint union, it suffices to show that i=1nAiL\bigcap_{i=1}^n A_i\in\msL, where each AiA_i is P\msP-compatible.

Without loss of generality, assume that A1c,,AmcPA_1^c,\ldots, A_m^c\in\msP and Am+1,,AnPA_{m+1},\ldots,A_n\in\msP. Let Bi=AicB_i=A_i^c and C=i=m+1nAiLC=\bigcap_{i=m+1}^n A_i\in\msL (not necessarily in P\msP: if m=nm=n then C=ΩC=\Omega). We have

i=1nAi=Ci=1mAi=C(i=1mAic)c=C(Ci=1mBi)=Ci=1mCBi.\bigcap_{i=1}^n A_i = C\cap\bigcap_{i=1}^m A_i = C\cap \left(\bigcup_{i=1}^m A_i^c\right)^c = C\setminus \left(C\cap \bigcup_{i=1}^m B_i \right) = C\setminus\bigcup_{i=1}^m C\cap B_i.

Note that each CBiC\cap B_i is in P\msP, even when C=ΩC=\Omega. In light of this last expression, we shall shift our goal to showing that finite union of P\msP-sets is in L\msL. For the case of two sets, if D1D_1 and D2D_2 are P\msP-sets, we have D1D2=D1(D2(D1D2))D_1\cup D_2 = D_1\cup (D_2\setminus (D_1\cap D_2)), a disjoint union of sets in L\msL. Suppose the union of any m1m-1 sets from P\msP lies in L\msL. Given D1,,DmPD_1,\ldots,D_m\in\msP, we have

i=1mDi=(D1i=2mDi)i=2mDi=(D1i=2mD1Di)i=2mDi.\bigcup_{i=1}^m D_i = \left(D_1\setminus \bigcup_{i=2}^m D_i\right)\cup \bigcup_{i=2}^m D_i = \left(D_1\setminus \bigcup_{i=2}^m D_1\cap D_i\right)\cup \bigcup_{i=2}^m D_i.

By the induction hypothesis, i=2mD1DiL\bigcup_{i=2}^m D_1\cap D_i\in\msL, so the above is a disjoint union of sets in L\msL. By induction, the result holds for all finite mm.

To recap, by showing that the finite union of P\msP-sets is in L\msL, we deduce the finite intersection of P\msP-compatible sets is in L\msL, it follows that L\msL contains the field f(P)f(\msP). But L\msL is a monotone class as well, so by the monotone class theorem, L\msL contains σ(f(P))=σ(P)\sigma(f(\msP))=\sigma(\msP), which gives us Dynkin’s theorem.

Footnotes

  1. This is theorem 3.1 in “Probability and Measure” by Billingsley and is a version of the Carathéodory’s extension theorem: a pre-measure on a given ring R\mcR of subsets of Ω\Omega can be extended to a measure on the σ\sigma-ring generated by R\mcR, and this extension is unique if the measure is σ\sigma-finite. The stated result follows from this general version, as a probability measure is both a pre-measure and a σ\sigma-finite measure.

  2. Intuitively, the tuples in MM gives us the indices for the terms of i=1mj=1niaij=(a11++a1n1)(am1++amnm)=jMi=1maiji\prod_{i=1}^m\sum_{j=1}^{n_i} a_{ij} = (a_{11}+\cdots+a_{1n_1})\cdots (a_{m1}+\cdots+a_{mn_m}) = \sum_{\textit{\textbf{j}}\in M}\prod_{i=1}^m a_{i\textit{\textbf{j}}_i}. Morphing sums into unions and products into intersections, we get the stated identity (a rigorous proof can be given by induction on mm).