It is often the case that two measures which agree on a certain class of sets actually agree on all sets in the relevant σ-algebra. There are a couple of standard tools to prove that the measures are the same: the Monotone Class Theorem and Dynkin’s π−λ theorem. These are essentially equivalent devices and it is largely a matter of taste which one to take as standard equipment.
Dynkin’s π-λ theorem is a powerful result in measure theory that is often used to extend properties of measures (or functions) defined on simple sets (like intervals or rectangles) to a larger σ-algebra, like the Borel σ-algebra. We first define \(\pi\) and \(\lambda\) systems before stating and proving the main theorem.
A π-system on a set \(\Omega\) is a non-empty collection \(\mathcal{P}\) of subsets of \(\Omega\) that is closed under finite intersections, that is \[ A,B\in P \implies A \cap B \in P. \]
The importance of \(\pi\)-systems arises from the fact that if two [[probability-measure|probability measures]] agree on a \(\pi\)-system, then they agree on the [[generated-sigma-algebra|\(\sigma\)-algebra generated]] by that \(\pi\)-system. Moreover, if other properties, such as equality of integrals, hold for the \(\pi\)-system, then they hold for the [[generated-sigma-algebra|generated \(\sigma\)-algebra]] as well. This is the case whenever the collection of subsets for which the property holds is a system.
A \(\lambda\)-system on \(\Omega\) is a set \(\mathcal{L}\) of subsets of \(\Omega\) satisfying: 1. \(\Omega \in \mathcal{L}\); 2. \(A,B \in \mathcal{L}~~\& ~~A \subset B \implies B \setminus A \in \mathcal{L}\); and 3. \(\{ A_{n} \}_{n=1}^\infty \in \mathcal{L}\) and \(A_{n}\uparrow A\) then \(A\in \mathcal{L}\).
Intuitively, we can think of the \(\pi\)-system representing a small, manageable collection of sets (like the real intervals, cylinder sets, rectangles etc.) where a property (e.g. equality of measures, independence etc.) is easily verified. The \(\lambda\)-system is designed to be stable under operations needed to build up to a \(\sigma\)-algebra (differences and countable increasing unions) but not necessarily intersections. The theorem states that once a property holds for the \(\pi\)-system, it propagates up to the \(\sigma\)-algebra generated by it, provided the \(\lambda\)-system contains the \(\pi\)-system.
If \(\mathcal{P}\) is a \(\pi\)-system, \(\mathcal{L}\) is a \(\lambda\)-system and \(\mathcal{P} \subset \mathcal{L}\) then \(\sigma(\mathcal{P})\subset \mathcal{L}\).
Proof: Step 1: We prove that if \(\mathcal{l}(\mathcal{P})\) is the smallest \(\lambda\)-system containing \(\mathcal{P}\), then \(\mathcal{l}(\mathcal{P})\) is a \(\sigma\)-algebra. Note that a \(\lambda\)-system that is closed under intersections is a \(\sigma\)-algebra since: (1) \(\Omega \in\mathcal{L}\); (2) \(A\in\mathcal{L}\implies A^c=\Omega \setminus A\in\mathcal{L}\) by second property of \(\lambda\)-systems; and (3) \(A \cup B = (A^c \cap B^c)^c\) and \(\bigcup_{i=1}^n A_{i}\uparrow\bigcup_{i=1}^\infty A_{i}\in\mathcal{L}\) by the third property of \(\lambda\)-systems.
Step 2: We show that \(\mathcal{l}(\mathcal{P})\) is closed under intersections. We define \(\mathcal{G}_{A}=\{ B:A \cap B \in \mathcal{l}(\mathcal{P}) \}\) and prove that if \(A \in \mathcal{l}(\mathcal{P})\), then \(\mathcal{G}_{A}\) is a \(\lambda\)-system.
Step 3: We assume \(A\in\mathcal{l}(\mathcal{P})\) and show that \(\mathcal{G}_{A}\) is a \(\lambda\)-system by noting that: (1) \(\Omega \in \mathcal{G}_{A}\) since \(A \in \mathcal{l}(\mathcal{P})\); (2) \(B,C \in \mathcal{G}_{A}\) and \(B \supset C\) \(\implies A \cap (B\setminus C)=(A \cap B)\setminus (A \cap C) \in\mathcal{l}(\mathcal{P})\) since \(A \cap B,~A \cap C \in\mathcal{l}(\mathcal{P})\); and (3) \(B_{n}\in \mathcal{G}_{A}\) and \(B_{n}\uparrow B\) \(\implies A \cap B_{n}\uparrow A \cap B \in \mathcal{l}(\mathcal{P})\) since \(A \cap B_{n}\in \mathcal{l}(\mathcal{P})\).
Note that \(\mathcal{P}\) is a \(\pi\)-system and so if \(A \in \mathcal{P}\) then \(\mathcal{G}_{A}\supset \mathcal{P}\), hence step (3) \(\implies \mathcal{G}_{A}\supset \mathcal{l}(\mathcal{P})\), i.e. if \(A \in \mathcal{P}\) and \(B\in \mathcal{l}(\mathcal{P})\) then \(A \cap B \in\mathcal{l}(\mathcal{P})\). Interchanging \(A\) and \(B\) in this sentence implies that if \(A \in \mathcal{l}(\mathcal{P})\) then \(\mathcal{G}_{A}\supset \mathcal{P}\) and so if \(A\in \mathcal{l}(\mathcal{P})\) then \(\mathcal{G}_{A}\supset \mathcal{P}\) and so step (3) \(\implies \mathcal{G}_{A}\supset \mathcal{l}(\mathcal{P})\).
This conclusion gives that if \(A,B\in \mathcal{l}(\mathcal{P})\) then \(A\cap B\in n\mathcal{l}(\mathcal{P})\), which proves step (2). Hence we have that \[
\sigma(\mathcal{P})\subset \mathcal{l}(\mathcal{P})\subset\mathcal{L},
\] giving the result. \(\square\)
Example: Uniqueness of Measures on \(\mathbb{R}\) We suppose that two measures \(\mu\) and \(\nu\) on \((\mathbb{R}, \mathcal{B}_{\mathbb{R}})\) agree on all intervals \(\{ (-\infty, x]:x \in \mathbb{R} \}\). We wish to show that \(\mu = \nu\) on all Borel sets.