Mathematics.

foundations of measure theory

σ-Algebras

Measure Theory60 minDifficulty7 out of 10

You should know: set operations and de morgans laws, power set

Overview

A σ-algebra (sigma-algebra) is a collection of subsets of a given set that is closed under complementation and countable unions. σ-algebras form the backbone of measure theory: they precisely identify which subsets of a space are 'measurable', making it possible to assign consistent sizes (measures) to those sets. Without the σ-algebra framework, paradoxes like the Banach–Tarski paradox show that naive notions of size lead to contradictions.

Intuition

Think of a σ-algebra as a 'universe of discourse' for a measuring device. If you can measure a set A, you should be able to measure its complement (everything outside A). If you can measure each set in a sequence A₁, A₂, …, you should be able to measure their union. The countability requirement (rather than arbitrary unions) is crucial: it prevents pathological constructions and allows limiting arguments central to analysis.

Formal Definition

Definition

Let X be a non-empty set. A σ-algebra on X is a collection ℱ of subsets of X satisfying three axioms.

X∈FX \in \mathcal{F}
Axiom 1 — contains the whole space
A∈F  ⟹  Ac∈FA \in \mathcal{F} \implies A^c \in \mathcal{F}
Axiom 2 — closed under complementation
A1,A2,…∈F  ⟹  ⋃n=1∞An∈FA_1, A_2, \ldots \in \mathcal{F} \implies \bigcup_{n=1}^{\infty} A_n \in \mathcal{F}
Axiom 3 — closed under countable unions
The pair (X,F) is called a measurable space.\text{The pair } (X, \mathcal{F}) \text{ is called a measurable space.}
Measurable space

Notation

NotationMeaning
F,  A\mathcal{F},\; \mathcal{A}Generic σ-algebra
B(R)\mathcal{B}(\mathbb{R})Borel σ-algebra on ℝ — generated by all open sets
σ(C)\sigma(\mathcal{C})σ-algebra generated by a collection 𝒞
(X,F)(X, \mathcal{F})Measurable space

Properties

Closed under countable intersections

A1,A2,…∈F  ⟹  ⋂n=1∞An∈FA_1, A_2, \ldots \in \mathcal{F} \implies \bigcap_{n=1}^{\infty} A_n \in \mathcal{F}

Condition: Follows from De Morgan's law and Axioms 2–3

Contains the empty set

∅∈F\emptyset \in \mathcal{F}

Condition: Since X ∈ ℱ and ∅ = Xᶜ

Closed under set differences

A,B∈F  ⟹  A∖B∈FA, B \in \mathcal{F} \implies A \setminus B \in \mathcal{F}

Condition: A \ B = A ∩ Bᶜ

Worked Examples

  1. The possible subsets of {a, b} are ∅, {a}, {b}, {a,b}.

    2X={∅,{a},{b},{a,b}}2^X = \{\emptyset,\{a\},\{b\},\{a,b\}\}

  2. The trivial σ-algebra {∅, X} satisfies all three axioms.

    F1={∅,{a,b}}\mathcal{F}_1 = \{\emptyset, \{a,b\}\}

  3. The power-set satisfies all axioms, giving a second σ-algebra.

    F2={∅,{a},{b},{a,b}}\mathcal{F}_2 = \{\emptyset, \{a\}, \{b\}, \{a,b\}\}

  4. Neither {∅, {a}} nor {∅, {b}} is closed under complementation (the complement of {a} is {b} which is missing), so they are not σ-algebras.

Answer: Exactly two σ-algebras: the trivial one {∅, {a,b}} and the full power set.

Practice Problems

Difficulty 6/10

Let X = {1, 2, 3}. Find all σ-algebras on X and count them.

Difficulty 7/10

Let 𝒞 be the collection of all open intervals (a, b) ⊆ ℝ. Show that σ(𝒞) = ℬ(ℝ).

Difficulty 8/10

Define the product σ-algebra on X × Y given σ-algebras ℱ on X and 𝒢 on Y, and show it is a σ-algebra.

Common Mistakes

Common Mistake

Confusing 'closed under finite unions' with 'closed under countable unions'

An algebra of sets is closed under finite unions; a σ-algebra requires closure under countable (possibly infinite) unions. The distinction is critical for Lebesgue measure.

Common Mistake

Thinking σ-algebras must be closed under uncountable unions

Closure is required only for countable unions. The Borel σ-algebra on ℝ contains singletons, but the uncountable union of all singletons in [0,1] is [0,1] — which is also Borel, but for a different reason.

Common Mistake

Assuming the power set 2^X is always the 'right' σ-algebra

2^X is too large in general: on uncountable sets like ℝ, consistent measures cannot be defined on 2^ℝ (Vitali sets). One works with a smaller σ-algebra such as ℬ(ℝ).

Quiz

Which of the following is NOT necessarily in a σ-algebra ℱ on X?
The σ-algebra generated by a collection 𝒞 is defined as:
True or False: Every algebra of sets is a σ-algebra.

Historical Background

The concept emerged in the early 20th century as mathematicians sought to rigorously extend the notion of length and area beyond intervals and rectangles. Henri Lebesgue's 1902 dissertation introduced what we now recognise as the Lebesgue integral, implicitly relying on σ-algebraic structure. Émile Borel formalised the smallest σ-algebra containing all open sets of the real line — the Borel σ-algebra — around the same time.

  1. 1902

    Lebesgue defines his integral, implicitly using σ-algebraic ideas

    Henri Lebesgue

  2. 1898

    Borel introduces the Borel σ-algebra on ℝ

    Émile Borel

  3. 1914

    Hausdorff's Grundzüge der Mengenlehre systematises set-theoretic foundations

    Felix Hausdorff

Summary

  • A σ-algebra on X is a collection of subsets containing X, closed under complementation and countable unions.
  • The pair (X, ℱ) is called a measurable space; elements of ℱ are called measurable sets.
  • The Borel σ-algebra ℬ(ℝ) is generated by the open sets of ℝ and contains all open, closed, Gδ, and Fσ sets.
  • The generated σ-algebra σ(𝒞) is the intersection of all σ-algebras containing 𝒞 — the smallest one doing so.
  • Arbitrary intersections of σ-algebras are σ-algebras, but arbitrary unions need not be.

References

  1. BookRudin, W. — Real and Complex Analysis, 3rd ed. (1987), Chapter 1
  2. BookFolland, G.B. — Real Analysis: Modern Techniques and Their Applications, 2nd ed. (1999), §1.1
  3. WebsiteWikipedia — Sigma-algebra