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
Recall that a Monotone Class is a family (i.e. class) \(M\) of sets that is closed under countable monotone unions and countable monotone intersections.
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 \(\mathcal{A}\).