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In this chapter, we would like to study, for a given set <math>\Omega</math>, subsets of the power set <math>\mathcal P(\Omega)</math>. We consider in particular those subsets of <math>\mathcal P(\Omega)</math> that are closed under certain operations.

{{definition|π-system|Let <math>\Omega</math> be a set. A '''<math>\pi</math>-system''' is a collection of sets <math>\mathcal A \subseteq \mathcal P(\Omega)</math> such that whenever <math>A, B \in \mathcal A</math>, then also <math>A \cap B \in \mathcal A</math>.}}

{{definition|Dynkin system|Let <math>\Omega</math> be a set. A '''Dynkin system''' or <math>\lambda</math>-system is a collection of sets <math>\Sigma \subseteq \Omega</math> such that the following three axioms hold:
#<math>\Omega \in \Sigma</math>
#<math>A, B \in \Sigma \Rightarrow A \setminus B \in \Sigma</math>
#<math>A_1, A_2, \cdots \in \Sigma \wedge A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots \Rightarrow \bigcup_{n \in \mathbb N} A_n \in \Sigma</math>.}}

{{definition|σ-algebra|Let <math>\Omega</math> be a set. A '''<math>\sigma</math>-algebra''' on <math>\Omega</math> is a collection of subsets of <math>\Omega</math>, say <math>\mathcal F \subseteq \mathcal P(\Omega)</math>, such that the following axioms are satisfied:
#<math>\Omega \in \mathcal F</math>
#<math>A, B \in \mathcal F \Rightarrow A \setminus B \in \mathcal F</math>
#<math>A_n \in \mathcal F</math> for all <math>n \in \mathbb N</math> implies <math>\bigcup_{n \in \mathbb N} A_n \in \mathcal F</math>.}}

Note that being a <math>\sigma</math>-algebra is a stronger requirement than being a Dynkin system: A <math>\sigma</math>-algebra is closed under all countable intersections, whereas a Dynkin system is only closed under intersections of countable ascending chains.

{{definition|σ-algebra generated by a collection of sets|Let <math>\Omega</math> be a set, and let <math>\mathcal A \subseteq 2^\Omega</math>. Then we define
:<math>\sigma(\mathcal A) := \bigcap_{\mathcal B \supset \mathcal A \atop \mathcal B \text{ is a } \sigma\text{-algebra}} \mathcal B</math>.}}

{{definition|λ-system generated by a collection of sets|Let <math>\Omega</math> be a set, and let <math>\mathcal A \subseteq 2^\Omega</math>. Then we define
:<math>\lambda(\mathcal A) := \bigcap_{\mathcal B \supset \mathcal A \atop \mathcal B \text{ is a } \lambda\text{-system}} \mathcal B</math>.}}

{{theorem|Dynkin's λ-π theorem|Let <math>\Omega</math> be a set, and let <math>\mathcal A</math> be a <math>\pi</math>-system on <math>\Omega</math>. Then
:<math>\sigma(\mathcal A) = \lambda(\mathcal A)</math>.}}

{{proof|The direction "<math>\supseteq</math>" is clear, so that we only have to prove "<math>\supseteq</math>". To do so, we prove that <math>\lambda(\mathcal A)</math> is in fact a <math>\sigma</math>-algebra that contains <math>\mathcal A</math>, using the definition of <math>\sigma(\mathcal A)</math> as the intersection of all <math>\sigma</math>-algebrae that contain <math>\mathcal A</math>.}}

== Exercises ==
#Let <math>\Omega</math> be a set, and let <math>\Sigma \subseteq \mathcal P(\Omega)</math>. Prove that <math>\Sigma</math> is a <math>\lambda</math>-system if and only if
##<math>\emptyset \in \Sigma</math>
##<math>A, B \in \Sigma \Rightarrow A \setminus B \in \Sigma</math>
##<math>A_1, A_2, \ldots \in \Sigma \wedge A_1 \supseteq A_2 \supseteq A_3 \supseteq \cdots \Rightarrow \bigcap_{n \in \mathbb N} A_n \in \Sigma</math>.
#Let <math>\Omega</math> be a set, and let <math>\mathcal F \subseteq \mathcal P(\Omega)</math>. Prove that <math>\mathcal F</math> is a <math>\sigma</math>-algebra if and only if
##<math>\emptyset \in \mathcal F</math>
##<math>A, B \in \mathcal F \Rightarrow A \setminus B \in \mathcal F</math>
##<math>A_n \in \mathcal F</math> for all <math>n \in \mathbb N</math> implies <math>\bigcap_{n \in \mathbb N} A_n \in \mathcal F</math>.

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