Dynkin’s lemma∗
gel†
2013-03-22 2:03:10
Dynkin’s lemma is a result in measure theory showing that the σ-algebra
generated by any given π-system on a set X coincides with the Dynkin system
generated the π-system. The result can be used to prove that measures are
uniquely determined by their values on π-systems generating the required σalgebra. For example, the Borel σ-algebra on R is generated by the π-system
of open intervals (a, b) for a < b and consequently the Lebesgue measure 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.
Proof. As A is a π-system, the set D1 ≡ {S ⊆ X : S ∩ T ∈ D(A) for every T ∈
A} contains A. We show that D1 is also a Dynkin system.
First, for every T ∈ A, X ∩ T = T ∈ A so X is in D1 . Second, if S1 ⊆ S2 are
in D1 and T ∈ A then (S2 \ S1 ) ∩ T = (S2 ∩ T ) \ (S1 ∩ T ) is in D(A) showing that
S2 \ S1 ∈ D1 . Finally. if Sn ∈ D1 is a sequence increasing to S ⊆ X and T ∈ A
then Sn ∩ T is a sequence in D(A) increasing to S ∩ T . As Dynkin systems are
closed under limits of increasing sequences this shows that S ∩ T ∈ D(A) and
therefore S ∈ D1 . So D1 is indeed a Dynkin system. In particular, D(A) ⊆ D1
and S ∩ T ∈ D(A) for all S ∈ D(A) and T ∈ A.
We now set D2 ≡ {S ⊆ X : S ∩ T ∈ D(A) for every T ∈ D(A)} which, as
shown above, contains A. Also, as in the argument above for D1 , D2 is a Dynkin
system. Therefore, D(A) is contained in D2 and it follows that S ∩ T ∈ D(A)
for any S, T ∈ D(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,
S c = X \ S ∈ D(A) for every S ∈ D(A) and, as it is also a π-system, this shows
that D(A)
Sn is an algebra of sets on X. Finally, choose any
S sequence An ∈ D(A).
Then, m=1 Am is a sequence in D(A) increasing to n An which, as D(A) is
∗ hDynkinsLemmai created: h2013-03-2i by: hgeli version: h41273i Privacy setting: h1i
hTheoremi h28A12i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1
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).
References
[1] David Williams, Probability with martingales, Cambridge Mathematical
Textbooks, Cambridge University Press, 1991.
2