Sigma-Algebras and Measurable Spaces

1.1 Algebras of Sets

Let $X$ be a set. A collection $\mathcal{A} \subseteq \mathcal{P}(X)$ is an algebra of sets if:

1. $X \in \mathcal{A}$.
2. $A \in \mathcal{A}$ implies $A^c \in \mathcal{A}$ (closed under complements).
3. $A, B \in \mathcal{A}$ implies $A \cup B \in \mathcal{A}$ (closed under finite unions).

From these axioms it follows that $\varnothing \in \mathcal{A}$, $\mathcal{A}$ is closed under finite intersections ($A \cap B = (A^c \cup B^c)^c$), and under set difference ($A \setminus B = A \cap B^c$).

1.2 Sigma-Algebras

An algebra $\mathcal{F}$ is called a $\sigma$-algebra (or sigma-algebra) if it is also closed under countable unions: if $\{A_n\}_{n=1}^{\infty} \subseteq \mathcal{F}$, then $\bigcup_{n=1}^{\infty} A_n \in \mathcal{F}$.

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

Proposition 1.1. A $\sigma$-algebra is also closed under countable intersections and countable complements:

$\bigcap_{n=1}^{\infty} A_n = \left(\bigcup_{n=1}^{\infty} A_n^c\right)^c$

1.3 Generated Sigma-Algebras

If $\mathcal{C} \subseteq \mathcal{P}(X)$ is any collection of subsets of $X$, the $\sigma$-algebra generated by $\mathcal{C}$, denoted $\sigma(\mathcal{C})$, is the smallest $\sigma$-algebra containing $\mathcal{C}$. It equals the intersection of all $\sigma$-algebras containing $\mathcal{C}$:

$\sigma(\mathcal{C}) = \bigcap\{\mathcal{F} : \mathcal{C} \subseteq \mathcal{F},\ \mathcal{F} \text{ is a } \sigma\text{-algebra}\}$

Definition. Let $X$ be a topological space with topology $\tau$. The Borel $\sigma$-algebra $\mathcal{B}(X)$ is $\sigma(\tau)$, the $\sigma$-algebra generated by the open sets. Elements of $\mathcal{B}(X)$ are called Borel sets.

Proposition 1.2. In $\mathbb{R}^n$, $\mathcal{B}(\mathbb{R}^n) = \sigma(\mathcal{O}) = \sigma(\mathcal{C}) = \sigma(\mathcal{K})$, where $\mathcal{O}$ is the collection of open sets, $\mathcal{C}$ is the collection of closed sets, and $\mathcal{K}$ is the collection of compact sets.

Proposition 1.3. $\mathcal{B}(\mathbb{R})$ contains all intervals: $(a, b)$, $[a, b]$, $(a, b]$, $[a, b)$ for $a, b \in \mathbb{R} \cup \{-\infty, +\infty\}$.

1.4 Examples

Example 1. For any set $X$, $\{\varnothing, X\}$ and $\mathcal{P}(X)$ are $\sigma$-algebras (the trivial and discrete $\sigma$-algebras).

Example 2. The countable-cocountable $\sigma$-algebra on $X$: $\mathcal{F} = \{A \subseteq X : A \text{ is countable or } A^c \text{ is countable}\}$.

Example 3. On $\mathbb{R}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ is generated by intervals of the form $(a, \infty)$ with $a \in \mathbb{R}$.