Section 3: The $\pi$-$\lambda$ Theorem

From Monotone Classes to the $\pi$-$\lambda$ Reformulation

The monotone class theorem is favored by analysts. It follows the analytic philosophy of using simpler sets as approximations of more complicated sets. This formalism is particularly useful in integration theory, where measurable functions are approximated by simple functions. Other important situations in which the monotone class theorem plays a central role include

• Constructing measures
• Extending properties of functions
• Approximating measurable functions
• Proving results about integrals

Conceptually, the monotone class theorem is an approximation principle: complicated objects are obtained as limits of simpler ones.

However, in many applications, especially in probability theory, the monotone class formalism can be unwieldy to apply directly, since it typically begins with an algebra of sets. In the mid-twentieth century, Dynkin introduced a different formalism, now called the $\pi$-$\lambda$ theorem, which is especially well suited to probabilistic arguments.

Rather than emphasizing approximation, the $\pi$-$\lambda$ formalism emphasizes logical closure. Dynkin separated the basic closure properties of a $\sigma$-algebra into two simpler structures:

• $\pi$-systems, which are closed under finite intersections
• $\lambda$-systems, which are closed under complements and countable disjoint unions

A key observation is that any collection of sets that is both a $\pi$-system and a $\lambda$-system is automatically a $\sigma$-algebra. This allows one to verify properties on a small generating $\pi$-system and extend them to the entire $\sigma$-algebra using the $\pi$-$\lambda$ theorem. Conceptually, the $\pi$-$\lambda$ theorem is an equality principle: once two measures (or probabilities) agree on a generating $\pi$-system, they agree on the whole $\sigma$-algebra.

Although the monotone class theorem and the $\pi$-$\lambda$ theorem are equivalent in logical strength, the $\pi$-$\lambda$ theorem is particularly effective for

• Proving uniqueness of measures
• Showing two measures coincide
• Verifying independence of random variables
• Checking equality of distributions

The rest of this section will focus on developing the $\pi$-$\lambda$ theorem.

Examples of $\pi$-Systems

Recall that a $\pi$-system is a collection $\mathcal{P}$ of subsets of a set $X$ that is closed under finite intersections: if $A,B \in \mathcal{P}$, then $A \cap B \in \mathcal{P}$.

Many of the most common generating families in measure theory and probability theory are naturally $\pi$-systems. We record several important examples.

• The collection of half-open intervals on $\R$.
• The collection of half-open rectangles on $\R^n$.
• The collection of cylinder sets on $\Omega^{\N}$.

Lemma 2.3: Intersections of $\pi$-systems

Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be $\pi$-systems. Then, $\mathcal{P}_1 \cap \mathcal{P}_2$ is a $\pi$-system.

The proof of Lemma 2.3 is routine and left as an exercise. Similar to algebras, $\sigma$-algebras, and monotone classes, we can define the $\pi$-system generated by a set. More precisely, if $\mathcal{S}$ is a family of sets on the space $\Omega$, we define $\pi(\mathcal{S})$ to be the intersection of all $\pi$-systems on $\Omega$ that contain $\mathcal{S}$. Lemma 2.3 guarantees that the $\pi$-system generated by a set is well defined.

An equivalent, more constructive, definition of a $\pi$-system generated by a set is given by \(\pi(\mathcal{S}) = \left\{\bigcap_{i=1}^n S_i: S_1,\dots,S_n \in \mathcal{S}\right\}\) i.e. the set of all finite intersections of elements in $\mathcal{S}$.

Lemma 2.4: Product of $\pi$-systems

Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be $\pi$-systems. Then, $\mathcal{P}_1 \times \mathcal{P}_2$ is a $\pi$-system.

Lemma 2.4 can be proved by using the set identity \((A \times B) \cap (C \times D) = (A \cap C) \times (B \cap D)\)

Lemma 2.5: Pre-images of $\pi$-systems are $\pi$-systems

Let $\mathcal{P}$ be a $\pi$-system on the space $Y$. If $f:X \to Y$ is a function, then \(f^{-1}(\mathcal{P}) = \{ f^{-1}(A) : A \in \mathcal{P} \}\) is a $\pi$-system.

Proof.

Let $A,B \in f^{-1}(\mathcal{P})$. Then, there exist $C,D \in \mathcal{P}$ such that $A = f^{-1}(C)$ and $B = f^{-1}(D)$. Using the set identity \(f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)\) we see that $A \cap B \in f^{-1}(\mathcal{P})$. $\square$

Lemma 2.4 and Lemma 2.5 can be used to show that half-open rectangles and cylinder sets are $\pi$-systems. Lemma 2.5 is needed since cylinder sets may be realized as pre-images of rectangles under coordinate projection maps.

Although $\pi$-systems capture stability under intersections, they do not, in general, enjoy closure under complements or unions. To develop a framework capable of extending properties from a generating $\pi$-system to an entire $\sigma$-algebra, we now introduce $\lambda$-systems, which emphasize closure under complements and countable disjoint unions.

$\lambda$-systems (Dynkin Systems)

A $\lambda$-system on a space $\Omega$ is a collection $\mathcal{D}$ of subsets of $\Omega$ such that

• $\Omega \in \mathcal{D}$
• $\mathcal{D}$ is closed under complements
• $\mathcal{D}$ is closed under countable pairwise disjoint unions.

We will explore concrete examples of $\lambda$-systems after introducing measures.

As with $\pi$-systems, $\sigma$-algebras, and monotone classes, one may define the $\lambda$-system generated by a collection of sets. If $\mathcal{S}$ is a family of subsets of $\Omega$, we denote by $\lambda(\mathcal{S})$ the smallest $\lambda$-system containing $\mathcal{S}$. By “smallest,” we mean

\[\lambda(\mathcal{S}) = \bigcap \{ \mathcal{D} : \mathcal{D} \text{ is a $\lambda$-system and } \mathcal{S} \subset \mathcal{D} \}.\]

This definition is well posed because arbitrary intersections of $\lambda$-systems are again $\lambda$-systems (the verification is straightforward and left as an exercise).

Proposition 2.6: $\lambda$-systems are Monotone Classes

Let $\mathcal{D}$ be a $\lambda$-system on the space $\Omega$. Then, $\mathcal{D}$ is a monotone class.

Proof.

Let $(A_n)$ be an increasing sequence of sets in $\mathcal{D}$ such that $A_n \nearrow A$. Define a new sequence $(B_n)$ such that \(B_1 = A_1,\quad B_n = A_n\setminus A_{n-1}\, \text{ for $n >1$}\) Since $A_{n-1} \subset A_n$, the sets $A_n^c$ and $A_{n-1}$ are disjoint. From the set identity \(A_n \setminus A_{n-1} = (A_n^c \sqcup A_{n-1})^c\) and the fact that $\mathcal{D}$ is closed under complements and disjoint unions, we conclude that $B_n \in \mathcal{D}$ for every $n$. Since $A = \bigsqcup_{n=1}^{\infty} B_n$, closure under countable disjoint unions in $\mathcal{D}$ implies $A \in \mathcal{D}$. Thus, $\mathcal{D}$ is closed under limits of increasing sequences.

To show that $\mathcal{D}$ is closed under limits of decreasing sequences, let $(A_n)$ be a decreasing sequence of sets in $\mathcal{D}$ such that $A_n \searrow A$. Since $\mathcal{D}$ is closed under complements, $(A_n^c)$ is an increasing sequence of sets in $\mathcal{D}$ such that $A_n^c \nearrow A^c$. Since we have already shown that $\mathcal{D}$ is closed under monotone increasing limits, $A^c \in \mathcal{D}$. By closure under complements, $A \in \mathcal{D}$. This shows that $\mathcal{D}$ is closed under limits of decreasing sequences. $\blacksquare$

It is important to note that all $\sigma$-algebras are also $\lambda$-systems. In this sense \(\sigma\text{-algebras} \subset \lambda\text{-systems} \subset \text{monotone classes}\) By combining $\pi$-systems and $\lambda$-systems, we realize Dynkin’s goal of characterizing $\sigma$-algebras.

Lemma 2.7: $\pi$-system + $\lambda$-system = $\sigma$-algebra

Suppose that $\mathcal{S}$ is a collection of subsets of $\Omega$ that is a $\pi$-system and $\lambda$-systems. Then, $\mathcal{S}$ is a $\sigma$-algebra.

Proof.

We need to check the three properties of a $\sigma$-algebra.

1. Since $\mathcal{S}$ is a $\lambda$-system, $\Omega \in \mathcal{S}$.
2. Since $\mathcal{S}$ is a $\lambda$-system, $\mathcal{S}$ is closed under complements.
3. Let $A,B \in \mathcal{S}$. Observe that $A \cup B = (A^c \cap B^c)^c$. Since $\mathcal{S}$ is closed under complements $A^c,B^c \in \mathcal{S}$. Since $\mathcal{S}$ is a $\pi$-system, $A^c \cap B^c \in \mathcal{S}$, Thus, $A \cup B = (A^c \cap B^c)^c \in \mathcal{S}$. This argument can be extended to show that $\mathcal{S}$ is closed under finite unions. Moreover, the set identity $A \setminus B = A \cap B^c$ shows that $\mathcal{S}$ is closed under relative complements. To show that $\mathcal{S}$ is closed under countable unions, let $(A_n)$ be a sequence of sets. Define a new sequence $(B_n)$ where $B_1 = A_1$ and $B_n = A_n \setminus \left(\bigcup_{i=1}^{n-1}A_i\right)$. Since $\mathcal{S}$ is closed under finite unions and relative complements $B_n \in \mathcal{S}$ for each $n \in \N$. By construction \(\bigcup_{n=1}^{\infty} A_n = \bigsqcup_{n=1}^{\infty} B_n \in \mathcal{S}\) Thus, $\mathcal{S}$ is a $\sigma$-algebra. $\blacksquare$

The preceding proposition shows that a collection of sets that is simultaneously a $\pi$-system and a $\lambda$-system must be a $\sigma$-algebra. Thus, to prove that a property verified on a generating $\pi$-system extends to the entire $\sigma$-algebra it generates, it suffices to construct a suitable $\lambda$-system containing that $\pi$-system and then show that it is also closed under intersections.

The $\pi$-$\lambda$ theorem accomplishes precisely this. It provides a powerful criterion for extending properties from a $\pi$-system to the $\sigma$-algebra it generates and forms the foundation for many uniqueness and independence results in probability theory.

$\pi$-$\lambda$ Theorem (Sierpinski-Dynkin, 1928)

Let $\mathcal{P}$ be a $\pi$-system on the space $\Omega$. Then, \(\sigma(\mathcal{P}) = \lambda(\mathcal{P})\)

Proof.

Any $\sigma$-algebra is automatically a $\lambda$-system, so \(\lambda(\mathcal{P}) \subset \sigma(\mathcal{P})\) To show the opposite inclusion, it suffices to show that $\lambda(\mathcal{P})$ is a $\sigma$-algebra containing $\mathcal{P}$. Let $\mathcal{D} = \lambda(\mathcal{P})$. Our goal will be to show that $\mathcal{D}$ is a $\pi$-system and then apply Lemma 2.7. To this end, define \(\mathcal{D}_A = \{B \subset \Omega: A \cap B \in \mathcal{D}\}\) where $A$ is a fixed subset of $\Omega$.

Claim 1: If $A \in \mathcal{P}$, then $\mathcal{P} \subset \mathcal{D}_A$

Let $B \in \mathcal{P}$. Since $\mathcal{P}$ is a $\pi$-system $A \cap B \in \mathcal{P}$. Hence, $\mathcal{P} \subset \mathcal{D}_A$. $\square$

Claim 2: If $A \in \mathcal{D}$, then $\mathcal{D}_A$ is a $\lambda$-system

Since $A \cap \Omega = A \in \mathcal{D}$, $\Omega \in \mathcal{D}_A$. Next, we will show that $\mathcal{D}_A$ is closed under complements. Let $B \in \mathcal{D}_A$, so $A \cap B \in \mathcal{D}$. Since $\mathcal{D}$ is a $\lambda$-system, it is closed under complements, so $A^c \in \mathcal{D}$. Consider the set identity \(A \cap B^c = A \setminus B = A \setminus (A \cap B)\) Moreover, \(A \setminus (A\cap B) = A \cap (A\cap B)^c = (A^c \cup (A \cap B))^c\) Because $A \cap B \subset A$, the sets $A^c$ and $A \cap B$ are disjoint. Therefore, \(A \cap B^c = (A^c \sqcup (A \cap B))^c\) The right hand side consists of elements in $\mathcal{D}$ combined using only the operations of disjoint union and complements. Hence, $A \cap B^c \in \mathcal{D}$ and $B^c \in \mathcal{D}_A$. Thus, $\mathcal{D}_A$ is closed under complements. To show that $\mathcal{D}_A$ is closed under countable pairwise disjoint unions, suppose that $(B_n)$ is a sequence of pairwise disjoint sets in $\mathcal{D}_A$. By definition, $A \cap B_n \in \mathcal{D}$ for all $n \in \N$. Using the distributive property of intersection over unions \(A \cap \left(\bigsqcup_{n=1}^{\infty} B_n \right) = \bigsqcup_{n=1}^{\infty} (A \cap B_n)\) The right hand side of the equality is an element of $\mathcal{D}$ by the closure properties of $\lambda$-systems. Therefore, $\mathcal{D}_A$ is closed under pairwise disjoint countable unions. Thus, $\mathcal{D}_A$ is a $\lambda$-system. $\square$

By Claim 1 and Claim 2, for each fixed $A \in \mathcal{P}$, $\mathcal{D}_A$ is a $\lambda$-system containing $\mathcal{P}$. By the definition of $\lambda(\mathcal{P})$, we have \(\mathcal{D} = \lambda(\mathcal{P}) \subset \mathcal{D}_A\) In other words, $\mathcal{D}$ is closed under intersection of elements in $\mathcal{D}$ with elements in $\mathcal{P}$. We now show that the intersection-closure with $\mathcal{P}$ can be extended to intersection-closure with all of $\mathcal{D}$.

Claim 3: $\mathcal{D}$ is closed under intersection.

Let $A \in \mathcal{D}$ and $B \in \mathcal{P}$. Since $\mathcal{D}$ is closed under intersection of elements in $\mathcal{D}$ with elements in $\mathcal{P}$, $A \cap B \in \mathcal{D}$. Hence, $B \in \mathcal{D}_A$. Since $B \in \mathcal{P}$ was arbitrary, we conclude $\mathcal{P} \subset \mathcal{D}_A$. By Claim 2, $D_A$ is a $\lambda$-system. Since, $\mathcal{D}_A$ is a $\lambda$-system containing $\mathcal{P}$, we have \(\mathcal{D} = \lambda(\mathcal{P}) \subset \mathcal{D}_A\) Hence, $\mathcal{D}$ is closed under intersections with the element $A$. Since $A \in \mathcal{D}$, was arbitrary it follows that $\mathcal{D}$ is closed under intersections. $\square$

By Claim 3, $\mathcal{D} = \lambda(\mathcal{P})$ is a $\pi$-system. $\lambda(\mathcal{P})$ is a $\pi$-system and a $\lambda$-system, so Lemma 2.7 implies $\lambda(\mathcal{P})$ is a $\sigma$-algebra. Since $\lambda(\mathcal{P})$ is a $\sigma$-algebra containing $\mathcal{P}$, we have \(\sigma(\mathcal{P}) \subset \lambda(\mathcal{P})\) Combining the inclusions, yields the proof. $\blacksquare$

The $\pi$–$\lambda$ theorem is one of the main engines behind uniqueness: if two measures agree on a generating $\pi$-system, they agree everywhere. In the next section, we turn to the complementary problem: existence—how to construct measures from simple geometric data. Caratheodory’s construction provides the existence mechanism, and $\pi$–$\lambda$ provides the uniqueness mechanism.