Sigma-Algebras and Measures - Core Definitions

DefinitionSigma-Algebra

Let $X$ be a non-empty set. A collection $\mathcal{F}$ of subsets of $X$ is called a sigma-algebra (or $\sigma$ -algebra) if it satisfies:

$X \in \mathcal{F}$
If $A \in \mathcal{F}$ , then $A^c \in \mathcal{F}$ (closed under complements)
If $A_1, A_2, \ldots \in \mathcal{F}$ , then $\bigcup_{i=1}^{\infty} A_i \in \mathcal{F}$ (closed under countable unions)

The pair $(X, \mathcal{F})$ is called a measurable space.

The concept of a sigma-algebra forms the foundation of modern measure theory. It provides the mathematical framework for rigorously defining measures and integration. The key requirement is closure under countable operations, which distinguishes sigma-algebras from simple algebras of sets.

ExampleBasic Sigma-Algebras

Consider a set $X$ . The following are sigma-algebras on $X$ :

Trivial sigma-algebra: $\mathcal{F} = \{\emptyset, X\}$
Power set: $\mathcal{F} = \mathcal{P}(X)$ , the collection of all subsets of $X$
Generated by a point: For $x \in X$ , let $\mathcal{F} = \{\emptyset, \{x\}, \{x\}^c, X\}$

The trivial sigma-algebra is the smallest, while the power set is the largest possible sigma-algebra on $X$ .

DefinitionMeasure

Let $(X, \mathcal{F})$ be a measurable space. A function $\mu: \mathcal{F} \to [0, \infty]$ is called a measure if:

$\mu(\emptyset) = 0$
For any countable collection $\{A_i\}_{i=1}^{\infty}$ of pairwise disjoint sets in $\mathcal{F}$ : $\mu\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \mu(A_i)$

This property is called countable additivity or $\sigma$ -additivity.

The triple $(X, \mathcal{F}, \mu)$ is called a measure space.

Remark

The requirement of countable additivity is crucial. It ensures that measures behave well with respect to limiting operations, which is essential for integration theory. Finite additivity alone would be insufficient for developing a robust theory of integration.

Key properties that follow from these definitions include monotonicity ( $A \subseteq B$ implies $\mu(A) \leq \mu(B)$ ) and subadditivity. These foundational concepts enable the construction of integration theories that generalize the Riemann integral and provide powerful tools for analysis and probability theory.