The aim of this post is to prove that a cummulative distribution function (CDF) uniquely determines the probability distribution. This is a well known fundamental result that is quite intuitive but requires some advanced techniques to prove. In this post, we provide a complete proof using Dynkin’s π-λ Theorem. The previous post on this topic covered basic of topology and measure theory. If you are unfamiliar with concepts like measure, open set, Borel algebra then please see the previous post.
The main result we want to prove is the following.
Theorem: Let 
Notice that we want to prove 
Dynkin’s π-λ Theorem
To state the theorem we will have to define two more things:
Definition (π-system): A collection of subsets 
It is trivial to verify that every algebra of subsets is a π-system.
Definition (λ-system): A collection of subsets
1. 
2. For any 
3. For any 
This definition looks quite similar to that of a σ-algebra. Similarly a λ-system is closed under complement. Why? Let 
Lemma 1: Every σ-algebra is both a π-system and a λ-system
Proof: Trivial
With these two definition we are ready to state the Dynkin’s π-λ Theorem.
Dynkin’s π-λ Theorem: If 
The proof of this theorem is short but tricky. We defer the proof for now to first demonstrate the usefulness of the above result.
Usefulness: The way we use Dynkin’s π-λ Theorem is by showing that the property holds for a π-system and which generates the σ-algebra of interest. Then we show that the collection of sets where the property holds is a λ-system. Dynkin’s π-λ Theorem then will show that the property holds for the entire σ-algebra.
CDF Uniquely Determines the Distribution
Theorem: Let 
Proof: Let 
Claim 1: Let ![T = \left\{ \left(-\infty, x\right] \mid x \in (-\infty, \infty] \right\}](http://s0.wp.com/latex.php?latex=T+%3D+%5Cleft%5C%7B+%5Cleft%28-%5Cinfty%2C+x%5Cright%5D+%5Cmid+x+%5Cin+%28-%5Cinfty%2C+%5Cinfty%5D+%5Cright%5C%7D&bg=ffffff&fg=404040&s=0&c=20201002)
Proof of Claim 1: The proof directly exploits the relation between closed rays and CDF. For any value of ![x \in (-\infty, \infty]](http://s0.wp.com/latex.php?latex=x+%5Cin+%28-%5Cinfty%2C+%5Cinfty%5D&bg=ffffff&fg=404040&s=0&c=20201002)
![P_1(g^{-1}((-\infty, x])) = F(x) = G(x) = P_2(g^{-1}((-\infty, x]))](http://s0.wp.com/latex.php?latex=P_1%28g%5E%7B-1%7D%28%28-%5Cinfty%2C+x%5D%29%29+%3D+F%28x%29+%3D+G%28x%29+%3D+P_2%28g%5E%7B-1%7D%28%28-%5Cinfty%2C+x%5D%29%29&bg=ffffff&fg=404040&s=0&c=20201002)
Hence, ![(-\infty, x] \in T](http://s0.wp.com/latex.php?latex=%28-%5Cinfty%2C+x%5D+%5Cin+T&bg=ffffff&fg=404040&s=0&c=20201002)
![\,\, (-\infty, x] \cap (-\infty, y] = (-\infty, \min(x,y)] \,\,](http://s0.wp.com/latex.php?latex=%5C%2C%5C%2C+%28-%5Cinfty%2C+x%5D+%5Ccap+%28-%5Cinfty%2C+y%5D+%3D+%28-%5Cinfty%2C+%5Cmin%28x%2Cy%29%5D+%5C%2C%5C%2C&bg=ffffff&fg=404040&s=0&c=20201002)
Great! so now we have a π-system where the desired result holds and that generates 
Claim 2:
Proof of Claim 2: We prove each required property of a λ-system below.
- From unitarity property of a probability measure we have
. Therefore,
.
- Let
and
. Let
then
and
are disjoint. Therefore, for any probability measure
we have
. Then
. Therefore,
.
- Let
be a sequence of increasing sets and
. For any probability measure
we have:
. We have
for every value of
. If two sequences have the same members then they have the same limit. Hence,
. Therefore,
.
. Therefore, 
then 
and 
we have 
. Then
. Therefore, 
.
. For any probability measure 
. We have 
for every value of 
. If two sequences have the same members then they have the same limit. Hence, 
. Therefore, Wrapping up: We showed that 
Proof of Dynkin’s π-λ Theorem
In order to proof this result we will make use of a couple of basic lemma:
Lemma 2: Intersection of λ-systems is a λ-system.
Proof: The proof is trivial. For example, let 
Lemma 3: Intersection of σ-algebra is a σ-algebra
Proof: Trivial to verify.
Lemma 4: If a collection of subsets of
Proof: Let
- As
is a λ-system therefore,
. Similarly, using the second property of a λ-system we have,
.
- Let
then
.
- Let
be a collection of subsets of
. Define a sequence of increasing sets as follows:
and
. Further,
. If somehow we could show that
then we can show that
. This would be interesting since we will show that every λ-system is a σ-algebra as we haven’t used the property of a π-system so far. However, as we will see we need the property to show
. We make an inductive argument:
. Assume
for all values of $k <= i$. Now
. As
is closed under complement (point 2 above) and intersection (S is a π-system) therefore,
. Hence, proved.
is a λ-system therefore, 
.
then 
.
be a collection of subsets of 
and 
. Further, 
. If somehow we could show that 
then we can show that 
. This would be interesting since we will show that every λ-system is a σ-algebra as we haven’t used the property of a π-system so far. However, as we will see we need the property to show 
. Assume 
for all values of $k <= i$. Now 
. As 
. Hence, proved.Dynkin’s π-λ Theorem: If 
Proof: Let 
Now every σ-algebra is a λ-system and as 
Now this is where the proof gets tricky. We first define a rather unintuitive set:
For any 
Claim 1: 
Proof of Claim 1: We prove each property of a λ-system below:
therefore,
- Let
and
. This implies
and
. Further,
.
Asis a λ-system therefore,
. From the definition of
this implies
.
- Let
be a countable collection of sets in
. This implies,
. As
is a λ-system therefore,
. This implies,
.
therefore, 
and 
. This implies 
and 
. Further, 
.
. From the definition of 
.
be a countable collection of sets in 
. As 
. This implies, 
.So far we have not use the fact that
Claim 2: 
Proof of Claim 2: As 
Claim 3: Define 
Proof of Claim 3: For 
Using Claim 3 we will now strengthen Claim 2.
Claim 4:
Proof of Claim 4: Let 
Claim 5:
Proof of Claim 5: For 
At this point, take a moment to observe the interesting pattern in Claim 2-5.
Wrapping Up: Claim 5 implies 
At the beginning of the proof, we showed 
Further Reading
If you are interested in learning more about probability theory then John Pike‘s lecture notes are a useful resource:
- Elementary Probability Theory, John Pike
- Probability Theory 1, John Pike
- Probability Theory 2, John Pike