Probability is built from small classes of sets: rectangles, cylinders, intervals, and algebras. Extension and monotone-class theorems are the tools that let identities proved on generators become identities on full \(\sigma\)-fields.
Premeasures and generated sigma-fields
One usually specifies a probability on an algebra, then extends it.
Definition. An algebra \(\mathcal A\) of subsets of \(\Omega\) is closed under complements and finite unions. A premeasure \(\mu_0:\mathcal A\to[0,\infty]\) is finitely additive and countably additive whenever the disjoint union remains in \(\mathcal A\). The generated \(\sigma\)-field is denoted \(\sigma(\mathcal A)\).
Carath?odory extension
Extension is the formal justification for constructing measures from generators.
Theorem. If \(\mu_0\) is a \(\sigma\)-finite premeasure on an algebra \(\mathcal A\), then there is a unique measure \(\mu\) on \(\sigma(\mathcal A)\) extending \(\mu_0\).
Proof. Define \[ \mu^\ast(E) =\inf\left\{\sum_{n=1}^{\infty}\mu_0(A_n): E\subset\bigcup_nA_n,\ A_n\in\mathcal A\right\}. \] This is an outer measure. A set \(E\) is Carath?odory measurable when \[ \mu^\ast(T)=\mu^\ast(T\cap E)+\mu^\ast(T\setminus E) \quad\text{for every }T\subset\Omega. \] Such sets form a sigma-field and the restriction of \(\mu^\ast\) is a complete measure. If \(A\in\mathcal A\), split every algebra cover of \(T\) by \(A\) and \(A^c\); finite additivity of \(\mu_0\) proves the criterion. Covering \(A\) by itself gives \(\mu^\ast(A)\le\mu_0(A)\), while countable additivity of the premeasure gives the reverse inequality for every algebra cover. Hence the restriction extends \(\mu_0\).
For uniqueness, first restrict to a set \(E_k\in\mathcal A\) of finite mass. Two extensions agree on the \(\pi\)-system \(\{A\cap E_k:A\in\mathcal A\}\), so the \(\pi\)-\(\lambda\) theorem makes them agree on \(\sigma(\mathcal A)\cap E_k\). Choose \(E_k\uparrow\Omega\) from sigma-finiteness and use continuity from below.
Example (constructing a distribution from its cdf). An increasing right-continuous \(F\) with \(F(-\infty)=0\), \(F(\infty)=1\) defines \[ \mu_0((a,b])=F(b)-F(a) \] on the algebra of finite disjoint unions of half-open intervals. Right-continuity supplies premeasure continuity at decreasing empty intersections, and Carath?odory extension produces the unique probability law with distribution function \(F\).
Monotone classes
The monotone-class theorem turns algebraic verification into measurable verification.
Proposition. If \(\mathcal A\) is an algebra and \(\mathcal M\) is a monotone class containing \(\mathcal A\), then \(\sigma(\mathcal A)\subseteq\mathcal M\).
Proof. Let \(\mathcal M_0\) be the smallest monotone class containing \(\mathcal A\). One checks that \(\mathcal M_0\) is itself an algebra: closure under complements and finite intersections is verified by fixing one set and applying monotone closure. An algebra that is also a monotone class is a \(\sigma\)-field, because countable unions are limits of increasing finite unions. Hence \(\mathcal M_0=\sigma(\mathcal A)\).
Theorem (functional monotone class). Let \(\mathcal H\) be a vector space of bounded real functions on \(\Omega\) that contains constants and is closed under bounded monotone pointwise limits. If \(\mathcal H\) contains the indicators of a \(\pi\)-system \(\mathcal P\), then it contains every bounded \(\sigma(\mathcal P)\)-measurable function.
Proof. Let \[ \mathcal D=\{A:\mathbf1_A\in\mathcal H\}. \] Because constants belong to \(\mathcal H\), \(X\in\mathcal D\); vector-space closure gives complements, and bounded monotone-limit closure gives disjoint countable unions. Thus \(\mathcal D\) is a Dynkin system containing \(\mathcal P\), hence \(\sigma(\mathcal P)\subset\mathcal D\). Therefore \(\mathcal H\) contains all measurable simple functions. Any bounded nonnegative measurable function is an increasing limit of bounded simple functions, and signed functions follow by positive and negative parts.
Remark. A useful proof pattern is: verify an expectation identity for indicators of a generating \(\pi\)-system, close it under linear combinations, then invoke the functional monotone-class theorem. This is how identities for conditional expectations, kernels, and independent random variables extend from rectangles to arbitrary bounded measurable tests.