Monotone Class Theorem
The monotone class theorem connects monotone classes and [[sigma-algebra|\(\sigma\)-algebra]]. The theorem generally states that the smallest monotone class containing an algebra of sets \(G\) is precisely the [[generated-sigma-algebra|smallest \(\sigma\)-algebra]] containing \(G\).
It is used as a type of transfinite induction to prove many theorems including [[fubinis-theorem]]. ### Monotone Class Theorem for Sets
A monotone class is a family (i.e. class) \(M\) of sets that is closed under countable monotone unions and under countable monotone intersections. Explicitly:
- \(A_{1}, A_{2}, \dots \in M\) and \(A_1\subseteq A_{2} \subseteq \dots\) then \(\bigcup_{i=1}^\infty A_{i}\in M\);
- \(B_{1}, B_{2}, \dots \in M\) and \(B_1\supseteq B_{2} \supseteq \dots\) then \(\bigcap_{i=1}^\infty B_{i}\in M\).
Let \(G\) be an algebra of sets and define \(M(G)\) to be the smallest monotone class containing \(G\). Then \(M(G)\) is precisely the [[sigma-algebra|\(\sigma\)-algebra]] generated by \(G\) \(\sigma(G)=M(G).\)
0.1 Monotone Class Theorem for Functions
Let \(\mathcal{A}\) be a [[pi-system|\(\pi\)-system]] that contains \(\Omega\) and let \(\mathcal{H}\) be a collection of functions from \(\Omega\) to \(\mathbb{R}\) with the following properties: 1. If \(A\in\mathcal{A}\) then \(\mathbb{1}_{A}\in \mathcal{H}\); 2. If \(f,g \in \mathcal{H}\) and \(c \in \mathbb{R}\) then \(f+g\in \mathcal{H}\) and \(cf \in \mathcal{H}\); and 3. If \(f_{n}\in \mathcal{H}\) is a sequence of non-negative functions that increase to a bounded function \(f\) then \(f \in\mathcal{H}\).
Then \(\mathcal{H}\) contains all bounded functions that are measurable with respect to \(\sigma(\mathcal{A})\) which is the [[generated-sigma-algebra|\(\sigma\)-algebra generated]] by \(\mathcal{A}\).