Dynkin System

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1 Definitions #

\(\mathcal{P}\) is a \(\pi\)-system if: \[A,B \in \mathcal{P} \Longrightarrow A\cap B \in \mathcal{P}\]

Example

\(\left\{ (-\infty,a]: a\in \RR \right\}\) is a \(\pi\)-system.

\(\left\{ (a,b]:-\infty<a\leq b<\infty \right\}\) is a \(\pi\)-system. Note that here \(a\) may equal to \(b\), so that \(\emptyset\) is also included, otherwise this is not a \(\pi\)-system.

\(\mathcal{L}\) is a \(\lambda\)-system if:

  1. \(\Omega\in \mathcal{L}\)
  2. If \(A,B\in \mathcal{L}\) and \(A \subset B\), then \(B \setminus A \in \mathcal{L}\)
  3. If \(A_1, A_2, \dots \in \mathcal{L}\) and \(A_i \uparrow A\), then \(A \in \mathcal{L}\)

Theorem: \(\mathcal{F}\) is both a \(\pi\)-system and a \(\lambda\)-system, if and only if \(\mathcal{F}\) is a \(\sigma\)-field.

Proof

(=>)

The first two conditions in the definition of \(\sigma\)-field is trivial. We only need to verify the third: If \(A_i \in \mathcal{F}\), for \(i=1,2,3, \dots \), then \(\bigcup_{i=1}^{\infty} A_i\in \mathcal{F}\). This follows by observing that:

\[\bigcup_{i=1}^{\infty} A_i = A_1 \cup (A_1 \cup A_2) \cup (A_1 \cup A_2 \cup A_3) \cup \dots \] \(A_1 \cup A_2 = \left( A_1^c \cap A_2^c \right)^c\), and using the definitions of \(\lambda\)-system and \(\pi\)-system.

(<=)

Since \(A\cap B = (A^c \cup B^c)^c\), by definition, it is a \(\pi\)-system.

The first and the third condition in the definition of \(\lambda\)-system is easy.

For the second condition, since \(B\setminus A = (\Omega \setminus A) \cap B\), by definition this concludes the proof.

2 Dynkin’s \(\pi\)-\(\lambda\) Theorem #

If \(\mathcal{P}\) is a \(\pi\)-system, \(\mathcal{L}\) is a \(\lambda\)-system, and \(\mathcal{P}\subset \mathcal{L}\), then \(\sigma(\mathcal{P}) \subset \mathcal{L}\).


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