Sigma-Algebras and Measurable Spaces

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 ($A \setminus B = A \cap B^c$).

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:

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

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 $a, b \in \mathbb{R} \cup \{-\infty, +\infty\}$.

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 $a \in \mathbb{R}$.

flowchart TD
    A[1_Sigma Algebras And Measurable Spaces] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Sigma-algebras formalise the notion of which events can be assigned probabilities. Think of a sigma-algebra as the collection of questions you are able to answer about a random experiment. The axioms, closure under complements and countable unions, ensure that if you can measure the probability of an event, you can also measure its complement and the probability that at least one of countably many events occurs. The Borel sigma-algebra on the real line is generated by open intervals and is the smallest sigma-algebra containing all open sets. Sigma-algebras also model information flow: a filtration is an increasing family of sigma-algebras representing what is known at each time.

• Confusing algebras with $\sigma$-algebras. An algebra is closed only under finite unions; a $\sigma$-algebra requires countable unions. The collection of finite and cofinite subsets of $\mathbb{N}$ is an algebra but not a $\sigma$-algebra.
• Thinking the Borel $\sigma$-algebra contains all subsets of $\mathbb{R}$. It does not. The existence of non-Lebesgue-measurable sets (using the axiom of choice) shows $\mathcal{P}(\mathbb{R})$ is strictly larger than $\mathcal{B}(\mathbb{R})$.
• Assuming generated $\sigma$-algebras are easy to describe explicitly. The Borel $\sigma$-algebra is enormous; there is no constructive way to list all its elements.
• Forgetting that $\sigma(\mathcal{C})$ is the intersection of all $\sigma$-algebras containing $\mathcal{C}$. This definition is useful but non-constructive; it does not tell us what the elements look like.

Problem 1. Show that the countable-cocountable $\sigma$-algebra on an uncountable set $X$ is not generated by a countable collection of subsets.

Solution. Suppose $\mathcal{C}$ is a countable collection generating $\mathcal{F}$. Then each $A \in \mathcal{F}$ is obtained from $\mathcal{C}$ by countably many set operations, so every non-cocountable set in $\mathcal{F}$ must be countable. But $\mathcal{C}$ itself is countable, so the $\sigma$-algebra it generates is at most the size of the continuum, while the cocountable $\sigma$-algebra on an uncountable $X$ contains uncountably many cocountable sets (each complement of a singleton). Contradiction. $\blacksquare$

Problem 2. Prove that if $\mathcal{F}$ is a $\sigma$-algebra and $\{A_n\} \subseteq \mathcal{F}$, then $\limsup_{n\to\infty} A_n = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k \in \mathcal{F}$.

Solution. For each $n$, $\bigcup_{k=n}^\infty A_k \in \mathcal{F}$ (countable union). The intersection of these sets is then also in $\mathcal{F}$ (countable intersection). So $\limsup A_n \in \mathcal{F}$. Similarly $\liminf A_n \in \mathcal{F}$. $\blacksquare$

• Probability theory: Kolmogorov’s axioms define a probability space $(\Omega, \mathcal{F}, P)$ where $\mathcal{F}$ is a $\sigma$-algebra of events. Filtrations (increasing families of $\sigma$-algebras) model information flow in stochastic processes.
• Statistics: Sufficient statistics are defined in terms of conditional expectations, which require $\sigma$-algebras generated by the statistic. The Lehmann-Scheffé theorem uses $\sigma$-algebras for minimal sufficient statistics.
• Ergodic theory: Invariant $\sigma$-algebras capture the long-term behaviour of dynamical systems. The ergodic theorem relates time averages to space averages via conditional expectation on the invariant $\sigma$-algebra.
• Economics: In financial mathematics, the $\sigma$-algebra generated by asset prices models the information available to traders, and martingale pricing uses filtrations.

• Measures: Measures are countably additive set functions defined on sigma-algebras, assigning sizes to measurable sets.
• Lebesgue Integration: Lebesgue integration is built on measures defined on sigma-algebras, generalising the Riemann integral.
• Continuity: Continuous functions generate Borel sigma-algebras, and continuity is a measurable property.
• Quantum Mechanics
• Graph Theory
• Classical Mechanics
• Electromagnetism
• Statistical Learning
• Statistical Mechanics