Sigma-Algebras and Measures - Main Theorem

TheoremMonotone Class Theorem

Let $\mathcal{A}$ be an algebra of subsets of $X$ . Let $\mathcal{M}$ be a monotone class containing $\mathcal{A}$ . Then $\mathcal{M}$ contains the sigma-algebra $\sigma(\mathcal{A})$ generated by $\mathcal{A}$ .

A collection $\mathcal{M}$ is a monotone class if:

Whenever $A_1 \subseteq A_2 \subseteq \cdots$ with $A_i \in \mathcal{M}$ , then $\bigcup_{i=1}^{\infty} A_i \in \mathcal{M}$
Whenever $A_1 \supseteq A_2 \supseteq \cdots$ with $A_i \in \mathcal{M}$ , then $\bigcap_{i=1}^{\infty} A_i \in \mathcal{M}$

The Monotone Class Theorem is a powerful tool for proving that two measures agree on a sigma-algebra. It reduces the problem to checking agreement on a simpler algebra that generates the sigma-algebra.

ExampleApplication to Uniqueness

Suppose $\mu$ and $\nu$ are two measures on $(X, \mathcal{F})$ . If they agree on an algebra $\mathcal{A}$ that generates $\mathcal{F}$ (i.e., $\sigma(\mathcal{A}) = \mathcal{F}$ ), and if there exists a sequence $X_n \in \mathcal{A}$ with $X_n \uparrow X$ and $\mu(X_n) = \nu(X_n) < \infty$ , then $\mu = \nu$ on $\mathcal{F}$ .

This is particularly useful when $\mathcal{A}$ is the collection of all finite unions of intervals, which generates the Borel sigma-algebra.

Remark

The theorem's power lies in its ability to extend results from algebras to sigma-algebras. An algebra $\mathcal{A}$ of sets satisfies:

$X \in \mathcal{A}$
If $A \in \mathcal{A}$ , then $A^c \in \mathcal{A}$
If $A, B \in \mathcal{A}$ , then $A \cup B \in \mathcal{A}$ (finite unions only)

The difference between algebras and sigma-algebras is that algebras only require closure under finite unions, while sigma-algebras require closure under countable unions.

The Monotone Class Theorem is instrumental in establishing uniqueness results for measures, particularly for extending measures from simple sets to more complex sigma-algebras. It bridges the gap between algebraic simplicity and the full power of sigma-algebras needed for measure theory.