Measure Theory 1.2: Sigma-Algebras Defined | Probability Primer

A measure μ is a function from a σ-algebra Σ over a set X to the extended real numbers [0, ∞]. To qualify as a measure, it must satisfy two essential conditions: (1) There exists at least one set with finite measure, ensuring the measure isn't trivial; (2) Countable additivity: For any countable collection of pairwise disjoint sets {Aₙ}, μ(∪Aₙ) = Σμ(Aₙ). A measure space is the triple (X, Σ, μ) combining these components. Special types include finite measures (μ(X) < ∞) and σ-finite measures (X can be expressed as a countable union of finite-measure sets).

A measurable space consists of a set equipped with a sigma-algebra, where the sets in the sigma-algebra are called measurable sets; a measure is a non-negative function on this sigma-algebra that assigns 'size' to sets, satisfying that the measure of the empty set is zero and that measures of pairwise disjoint sets add up, enabling rigorous definitions of length, area, and volume for 'sensible' sets like Borel sets while avoiding pathological cases.

In mathematical finance, a sigma-algebra is a collection of subsets of a set X that includes the empty set, is closed under countable unions, and contains the complement of every set in the collection; a measure is a function from a sigma-algebra to non-negative real numbers that is non-negative, assigns zero to the empty set, and satisfies sigma-additivity (countable additivity); and a measure space is a triplet consisting of a set X, a sigma-algebra on X, and a measure defined on that sigma-algebra, forming the foundational framework for probability theory and financial derivative pricing.

A random variable is a measurable function from a measure space (Omega, F) to the real line with the Borel sigma-algebra. A function is measurable if the inverse image of any measurable set in the target space is measurable in the domain. Measurable functions serve as morphisms in measure theory, analogous to homeomorphisms in topology and group homomorphisms in group theory. To prove a function is a random variable, one must verify it maps the sample space to real numbers and that the inverse image of any Borel set is measurable in the domain sigma-algebra.

This section introduces random variables as measurable functions from a probability space to a measurable space (typically the real numbers with Borel sigma algebra). A function f: X → W is measurable if for every event B in the target sigma algebra, the preimage f⁻¹(B) is in the source sigma algebra. This measurability condition ensures that we can compute probabilities for events defined by the random variable, such as P(X ≥ c). Random variables allow us to assign numerical values to outcomes in a way that respects the probabilistic structure.

A function ξ: Ω → ℝ is measurable with respect to σ-algebra F if for every real c, the set {ω : ξ(ω) ≤ c} belongs to F. This ensures we can assign probabilities to statements about function values. The class of measurable functions depends on σ-algebra richness—richer σ-algebras make more functions measurable. For finite σ-algebras generated by partitions, measurable functions are constant on each partition element (simple functions). A random variable is any measurable function from (Ω, F, P) to ℝ. This generalizes the discrete case where any function was automatically measurable.

This section establishes the foundational concepts of measure-theoretic probability. A function between measurable spaces (Ω₁, F₁) and (Ω₂, F₂) is measurable if the preimage of any set in F₂ belongs to F₁. When Ω₂ = ℝ, we assume the Borel sigma field unless specified otherwise. A random variable is defined as a measurable real-valued function from a probability space. The key characterization theorem states that X: Ω → ℝ is a random variable if and only if for every real x, the set {ω | X(ω) ≤ x} is in F. This theorem provides a practical criterion for verifying measurability without checking all Borel sets.

A random variable is a measurable function with respect to the sigma field F. The term 'measurable' means the inverse image of any semi-closed interval (-∞, x] belongs to F. In advanced probability courses, random variables are called measurable functions. The definition requires only Ω and F; the probability function P is not needed for defining X.

A probability space consists of three components: (1) Ω (sample space) containing all elementary outcomes, (2) F (sigma-algebra) of events as subsets of Ω, and (3) P (probability function) assigning values in [0,1] to events. Elementary events have three key properties: mutual exclusivity (cannot occur simultaneously), indivisibility (cannot be divided into smaller meaningful parts), and completeness (cover the entire sample space). A sigma-algebra is a collection of subsets closed under complement, countable unions, and containing Ω. This structure ensures we can perform all necessary operations on events. The probability function must satisfy three axioms: non-negativity (P(A) ≥ 0), normalization (P(Ω) = 1), and countable additivity (P(∪Aᵢ) = ΣP(Aᵢ) for mutually exclusive events).

Kolmogorov's axiomatic system, formulated in 1933, provides a rigorous mathematical foundation for probability theory by defining a probability space as a triple (Ω, Σ, P), where Ω is the sample space of all possible outcomes, Σ is a σ-algebra of events (subsets of Ω closed under complementation and countable unions), and P is a probability measure satisfying three axioms: non-negativity (P(A) ≥ 0), normalization (P(Ω) = 1), and countable additivity (P(∪Aₙ) = ΣP(Aₙ) for pairwise disjoint events). This abstract framework enables the derivation of fundamental properties such as the complement rule (P(Aᶜ) = 1 - P(A)), monotonicity, and the addition principle for unions of events, forming the basis for both discrete probability spaces (with countable sample spaces) and continuous probability spaces (using Borel σ-algebras and probability densities).

The axiomatic definition of probability, formulated by Kolmogorov, defines probability as a function P on a sigma-algebra of events satisfying three axioms: non-negativity (P(A) ≥ 0), normalization (P(Ω) = 1), and countable additivity (P(∪Aₙ) = ΣP(Aₙ) for disjoint events). A probability space consists of a sample space Ω, a sigma-algebra F of events, and the probability measure P defined on F. Conditional probability P(A|B) = P(A∩B)/P(B) for P(B) > 0, and events A and B are independent if P(A∩B) = P(A)P(B).

This section introduces Kolmogorov's axiomatic approach to probability theory, which provides a formal mathematical framework for understanding probability. Events must form a closed structure closed under intersection, union, and complementation. The probability space consists of three components: Omega (sample space of elementary outcomes), F (collection of events), and P (probability measure). The axioms include: Omega is an event, closure under complementation, closure under finite union, and closure under countable union. The collection F is called a sigma-algebra. A crucial point is that if A is an event, its subsets need not be events (illustrated by geometric probability). The trivial sigma-algebra contains only ∅ and Ω, while the power set 2^Ω is a sigma-algebra when Ω is finite.

The Kolmogorov axioms, established by Russian mathematician Andrey Kolmogorov (1903-1987), provide the rigorous mathematical foundation for probability theory through three fundamental rules: (1) Non-negativity: probability of any event is ≥ 0, (2) Normalization: probability of the entire sample space equals 1, and (3) Countable additivity: probability of the union of mutually exclusive events equals the sum of their individual probabilities. These axioms define a probability space as a triple (Ω, F, P), where Ω is the sample space, F is a sigma-algebra of events, and P is the probability measure. This axiomatic framework enables mathematicians to model real-world random phenomena by constructing appropriate sample spaces and defining probability measures that satisfy these fundamental properties.

A sigma algebra generated by a class is the intersection of all sigma algebras containing that class. A class is a collection of non-empty sets and subsets of a universal set X. The sigma algebra generated by a class is the smallest sigma algebra that contains all sets in the class. For any class of subsets of X, there exists a smallest sigma algebra containing it, which is the intersection of all sigma algebras that contain the class. This intersection itself forms a sigma algebra and contains the original class.

For any collection A of subsets of X, there exists a unique smallest sigma algebra containing A. This sigma algebra is the intersection of all sigma algebras that contain A. It is analogous to the subspace generated by a set in linear algebra. The intersection is non-empty because the power set of X is a sigma algebra containing A. The generated sigma algebra is minimal and unique, allowing the notation σ(A) for the generated sigma algebra.

A sigma-algebra of subsets of a set X is a collection that includes the empty set and the whole space, is closed under complements, and is closed under countable unions. Every sigma-algebra is also an algebra, but not every algebra is a sigma-algebra. Given any class C of subsets of X, the sigma-algebra generated by C is defined as the intersection of all sigma-algebras that contain C, and this intersection itself forms the smallest sigma-algebra containing C.

This section introduces the Borel sigma algebra as the most important specific case of generated sigma algebras. The Borel sigma algebra on the real line is defined as the sigma algebra generated by open sets, which is also the smallest sigma algebra containing all open or closed intervals. The extended real line (real line plus plus/minus infinity) has a Borel sigma algebra consisting of Borel sets on the real line with infinity points added. Six different classes of subsets can generate the Borel sigma algebra: closed sets, open sets, open intervals, finite intervals (a,b], semi-infinite intervals [a,∞), and semi-infinite intervals (-∞,b). All these classes generate the same sigma algebra, demonstrating the flexibility in constructing Borel sets from different starting collections.