σ-algebras, π/λ-systems, and the monotone class theorem

2026-01-31

These notes collect a few standard “reference lemmas” that I keep reusing in later probability/stochastic notes.

Definition of a σ-algebra

Let $\Omega$ be a set. A σ-algebra $\mathcal F \subseteq \mathcal P(\Omega)$ is a collection of subsets of $\Omega$ such that:

$\Omega \in \mathcal F$.
If $A \in \mathcal F$, then $A^c := \Omega \setminus A \in \mathcal F$.
If $A_1, A_2, \dots \in \mathcal F$, then $\bigcup_{n\ge 1} A_n \in \mathcal F$.

Closure under countable intersections follows from De Morgan:

$$ \bigcap_{n\ge 1} A_n \;=\; \left(\bigcup_{n\ge 1} A_n^c\right)^c \in \mathcal F. $$

Definition of measurable space

A measurable space is a pair $(\Omega,\mathcal F)$ where $\Omega$ is a set and $\mathcal F$ is a σ-algebra on $\Omega$ (so the elements of $\mathcal F$ are called measurable sets).

Measurable map (measurable function)

Let $(E,\Sigma_1)$ and $(F,\Sigma_2)$ be measurable spaces, and let $f:E\to F$ be a function.

We say $f$ is $(\Sigma_1,\Sigma_2)$-measurable (or simply measurable) if

$$ \forall B\in\Sigma_2,\quad f^{-1}(B)\in \Sigma_1. $$

Equivalently: the preimage of every measurable set in $F$ is measurable in $E$.

Given measurable spaces $(\Omega_1,\mathcal F_1)$ and $(\Omega_2,\mathcal F_2)$, their product measurable space is

$$ (\Omega_1\times \Omega_2,\; \mathcal F_1 \otimes \mathcal F_2), $$

where $\mathcal F_1 \otimes \mathcal F_2$ is the product σ-algebra generated by measurable rectangles $A_1\times A_2$ with $A_1\in\mathcal F_1$ and $A_2\in\mathcal F_2$.

π-system and λ-system (Dynkin system)

π-system

A nonempty collection $\mathcal P \subseteq \mathcal P(\Omega)$ is a π-system if it is closed under finite intersections:

$$ A,B\in\mathcal P \implies A\cap B \in \mathcal P. $$

λ-system (Dynkin system)

A nonempty collection $\mathcal D \subseteq \mathcal P(\Omega)$ is a λ-system (a.k.a. Dynkin system) if:

$\Omega \in \mathcal D$.
If $A,B \in \mathcal D$ and $A\subseteq B$, then $B\setminus A \in \mathcal D$.
If $A_1,A_2,\dots \in \mathcal D$ are pairwise disjoint, then $\bigcup_{n\ge 1} A_n \in \mathcal D$.

Basic fact we use constantly

Every σ-algebra is a λ-system.
- Proof sketch: take $\Omega \in \mathcal F$. If $A\subseteq B$ and $A,B\in\mathcal F$, then $B\setminus A = B\cap A^c\in\mathcal F$. Disjoint countable union closure is a special case of countable union closure.
A λ-system is closed under complements.
- If $A\in\mathcal D$, then $A^c = \Omega\setminus A \in \mathcal D$ by property (2) with $A\subseteq \Omega$.
If $\mathcal D$ is a λ-system and $A,B\in\mathcal D$ with $A\subseteq B$, then $B\setminus A\in\mathcal D$ (this is literally the defining property).

Monotone class theorem (π-λ / Dynkin’s lemma)

There are two closely related “monotone class theorems” in common use. The one that pairs naturally with π/λ-systems is the following (often called the π-λ theorem).

Theorem (π-λ / Dynkin’s lemma)

Let $\mathcal P$ be a π-system on $\Omega$, and let $\mathcal D$ be a λ-system on $\Omega$ with $\mathcal P \subseteq \mathcal D$. Then

$$ \sigma(\mathcal P)\subseteq \mathcal D. $$

Equivalently, the smallest λ-system containing $\mathcal P$ contains $\sigma(\mathcal P)$.

Proof

Define

$$ \mathcal L := \bigcap\{\mathcal D' : \mathcal D' \text{ is a λ-system and } \mathcal P\subseteq \mathcal D'\}. $$

Then $\mathcal L$ is a λ-system (intersection of λ-systems is a λ-system), and $\mathcal L$ is the smallest λ-system containing $\mathcal P$.

Step 1: show $\mathcal L$ is a σ-algebra.
We already know $\mathcal L$ is a λ-system. To upgrade it to a σ-algebra, it suffices to show closure under finite intersections (then λ-system + π-system-like closure gives σ-algebra; a standard route is to show $\mathcal L$ is also a π-system).

Fix $A\in\mathcal P$. Consider the class

$$ \mathcal L_A := \{B\in\mathcal L : A\cap B \in \mathcal L\}. $$

We claim $\mathcal L_A$ is a λ-system:

$\Omega \in \mathcal L_A$ because $A\cap\Omega = A \in \mathcal P\subseteq \mathcal L$.
If $B,C\in\mathcal L_A$ with $B\subseteq C$, then
$$ A\cap (C\setminus B) = (A\cap C)\setminus (A\cap B)\in\mathcal L $$
since $A\cap C, A\cap B \in \mathcal L$ and $\mathcal L$ is a λ-system (closed under set difference when one is contained in the other). Hence $C\setminus B\in\mathcal L_A$.
If $B_1,B_2,\dots\in\mathcal L_A$ are pairwise disjoint, then
$$ A\cap \left(\bigcup_{n\ge 1} B_n\right) = \bigcup_{n\ge 1} (A\cap B_n) $$
is a disjoint union of sets in $\mathcal L$, so it lies in $\mathcal L$. Hence $\bigcup_{n\ge 1} B_n \in \mathcal L_A$.

Now check that $\mathcal P \subseteq \mathcal L_A$: if $B\in\mathcal P$, then $A\cap B\in\mathcal P$ (π-system property), so $A\cap B \in \mathcal P\subseteq \mathcal L$, hence $B\in\mathcal L_A$.

Since $\mathcal L$ is the smallest λ-system containing $\mathcal P$, and $\mathcal L_A$ is a λ-system containing $\mathcal P$, we must have $\mathcal L \subseteq \mathcal L_A$. Therefore:

$$ \forall B\in\mathcal L,\quad A\cap B \in \mathcal L. $$

So for each fixed $A\in\mathcal P$, $\mathcal L$ is closed under intersection with $A$.

Next fix $B\in\mathcal L$ and define

$$ \mathcal M_B := \{A\in\mathcal L : A\cap B \in \mathcal L\}. $$

The same argument as above shows $\mathcal M_B$ is a λ-system, and it contains $\mathcal P$ (because we just proved: for $A\in\mathcal P\subseteq\mathcal L$, $A\cap B \in \mathcal L$). Hence $\mathcal L\subseteq \mathcal M_B$, so in particular $A\cap B\in\mathcal L$ for all $A\in\mathcal L$.

Thus $\mathcal L$ is closed under finite intersections, i.e. $\mathcal L$ is a π-system.

Finally, a λ-system that is also a π-system is a σ-algebra: indeed, for any $A,B\in\mathcal L$,

$$ A\cup B = \left((A^c\cap B^c)\right)^c $$

and using complements (λ-system) + intersections (π-system) shows closure under finite unions; then finite unions + disjointification gives countable unions. Concretely: for any $A_1,A_2,\dots\in\mathcal L$, define disjoint $C_1=A_1$, $C_n=A_n\setminus\bigcup_{k\lt n}A_k$; each $C_n\in\mathcal L$ by repeated use of “difference” and finite unions, and $\bigcup_n A_n=\bigcup_n C_n\in\mathcal L$ by disjoint union closure. Hence $\mathcal L$ is a σ-algebra.

Step 2: conclude $\sigma(\mathcal P)\subseteq \mathcal L\subseteq \mathcal D$.
By construction $\mathcal P\subseteq \mathcal L$, and $\mathcal L$ is a σ-algebra, so $\sigma(\mathcal P)\subseteq \mathcal L$. Since $\mathcal L\subseteq \mathcal D$ for any λ-system $\mathcal D$ containing $\mathcal P$, we get $\sigma(\mathcal P)\subseteq\mathcal D$. ∎

These theorems are very fundamental, just keep in mind

To prove an identity/inequality that is known on a generating class $\mathcal P$ (often a π-system), you:

Define a class $\mathcal D$ of sets on which the statement holds.
Prove $\mathcal D$ is a λ-system.
Check $\mathcal P\subseteq \mathcal D$.
Conclude $\sigma(\mathcal P)\subseteq \mathcal D$, hence the statement holds on all of $\sigma(\mathcal P)$.