Dynkin’s lemma

Dynkin’s lemma is a result in measure theory showing that the σ-algebra (http://planetmath.org/SigmaAlgebra) generated by any given π-system (http://planetmath.org/PiSystem) on a set X coincides with the Dynkin system generated the π-system. The result can be used to prove that measuresMathworldPlanetmath are uniquely determined by their values on π-systems generating the required σ-algebra. For example, the Borel σ-algebra on is generated by the π-system of open intervals (a,b) for a<b and consequently the Lebesgue measureMathworldPlanetmath μ is uniquely determined by the property that μ((a,b))=b-a.

Note that this lemma generalizes the statement that a Dynkin system which is also a π-system is a σ-algebra.

Lemma (Dynkin).

Let A be a π-system on a set X. Then D(A)=σ(A). That is, the smallest Dynkin system containing A coincides with the σ-algebra generated by A.


As A is a π-system, the set 𝒟1{SX:ST𝒟(A) for every TA} contains A. We show that 𝒟1 is also a Dynkin system.

First, for every TA, XT=TA so X is in 𝒟1. Second, if S1S2 are in 𝒟1 and TA then (S2S1)T=(S2T)(S1T) is in 𝒟(A) showing that S2S1𝒟1. Finally. if Sn𝒟1 is a sequence increasing to SX and TA then SnT is a sequence in 𝒟(A) increasing to ST. As Dynkin systems are closed under limits of increasing sequences this shows that ST𝒟(A) and therefore S𝒟1. So 𝒟1 is indeed a Dynkin system. In particular, 𝒟(A)𝒟1 and ST𝒟(A) for all S𝒟(A) and TA.

We now set 𝒟2{SX:ST𝒟(A) for every T𝒟(A)} which, as shown above, contains A. Also, as in the argumentMathworldPlanetmathPlanetmath above for 𝒟1, 𝒟2 is a Dynkin system. Therefore, 𝒟(A) is contained in 𝒟2 and it follows that ST𝒟(A) for any S,T𝒟(A). So 𝒟(A) is both a π-system and a Dynkin system.

We can now show that 𝒟(A) is a σ-algebra. As it is a Dynkin system, Sc=XS𝒟(A) for every S𝒟(A) and, as it is also a π-system, this shows that 𝒟(A) is an algebra of setsMathworldPlanetmath on X. Finally, choose any sequence An𝒟(A). Then, m=1nAm is a sequence in 𝒟(A) increasing to nAn which, as 𝒟(A) is Dynkin system, must be in 𝒟(A). So, 𝒟(A) is a σ-algebra and must contain σ(A). Conversely, as σ(A) is a Dynkin system (as it is a σ-algebra) containing A, it must also contain 𝒟(A). ∎


  • 1 David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, 1991.
Title Dynkin’s lemma
Canonical name DynkinsLemma
Date of creation 2013-03-22 18:33:05
Last modified on 2013-03-22 18:33:05
Owner gel (22282)
Last modified by gel (22282)
Numerical id 11
Author gel (22282)
Entry type Theorem
Classification msc 28A12
Synonym pi-system d-system lemma
Related topic PiSystem
Related topic DynkinSystem
Related topic UniquenessOfMeasuresExtendedFromAPiSystem