σ-Algebras and Probability Spaces

Setup

Why measure theory?

Classical probability assigns a number in $[0,1]$ to each "event." On a finite sample space — six faces of a die, for instance — this is straightforward: you can attach a probability to every subset. The trouble starts when the sample space is uncountable, like the real line or the set of continuous asset price paths.

It turns out to be impossible to assign a translation-invariant, countably additive measure extending length to every subset of $\mathbb{R}$. The Vitali construction (which uses the Axiom of Choice) produces a set to which no consistent measure can be assigned. The resolution is not to measure every subset, but to choose a σ-algebra $\mathcal{F}$ — a designated collection of admissible subsets of $\mathbb{R}$. The elements of $\mathcal{F}$ are then called measurable sets. For Lebesgue measure, $\mathcal{F}$ is the Lebesgue σ-algebra, which contains all Borel sets and many more, but not every subset of $\mathbb{R}$.

Conventions used throughout

• $\Omega$ denotes a non-empty set (the sample space — the collection of all possible outcomes).
• $A^c = \Omega \setminus A$ denotes the complement of $A$ in $\Omega$.
• "Countable" means finite or countably infinite.
• Indices for countable unions and intersections run over $\mathbb{N} = \{1, 2, 3, \ldots\}$ unless otherwise stated.

Theory

1. σ-Algebras

The three axioms are not arbitrary. Axiom 1 ensures the certain event is trackable. Axiom 2 ensures "event does not occur" is trackable whenever "event occurs" is. Axiom 3 — requiring countable rather than merely finite closure — is the key restriction: it is strong enough to support Lebesgue measure on $\mathbb{R}$, but weak enough to exclude the pathological subsets.

These are not additional axioms — they are derivable. Many students list "$\emptyset \in \mathcal{F}$" as an axiom; it is not.

Two canonical examples

The trivial σ-algebra $\{\emptyset, \Omega\}$ is the coarsest possible: it carries no information about which outcome actually occurred. The power set $2^\Omega$ (all subsets) is the finest: every event is trackable. For uncountable $\Omega$ — such as $\mathbb{R}$ — the power set is too large to support a consistent measure, which is precisely why we need something in between.

Generated σ-algebras

Given any collection $\mathcal{C}$ of subsets of $\Omega$, the σ-algebra generated by $\mathcal{C}$ is the smallest σ-algebra containing every set in $\mathcal{C}$:

This is well-defined because the power set $2^\Omega$ always works as a bounding σ-algebra, so the intersection is over a non-empty family.

The generating family $\{(-\infty, x]\}$ is particularly important: it connects $\mathcal{B}(\mathbb{R})$ directly to the cumulative distribution function, since $F_X(x) = \mathbb{P}(X \le x) = \mathbb{P}(X^{-1}((-\infty, x]))$ is the preimage of a Borel generator.

2. Probability Measures

σ-additivity (countable, not just finite additivity) implies that $\mathbb{P}$ is continuous: if $A_n \nearrow A$ then $\mathbb{P}(A_n) \nearrow \mathbb{P}(A)$. This continuity is what enables all standard limit theorems used in pricing.

3. Random Variables

Measurability says: for every Borel region $B$ on the real line, the event "the outcome lands in the preimage of $B$" is trackable in our σ-algebra. A non-measurable $X$ would mean some regions of $\mathbb{R}$ have preimages outside $\mathcal{F}$ — we could not assign them probabilities.

4. Geometric intuition

Think of $\Omega$ as a large bag containing all possible "states of the world." A σ-algebra is the list of questions you are permitted to ask about which state actually occurred.

• The trivial σ-algebra $\{\emptyset, \Omega\}$ allows only: "did something happen?" — it carries zero information.
• The power set $2^\Omega$ allows every conceivable question — but on uncountable $\Omega$, assigning consistent probabilities to all answers is impossible.
• $\mathcal{B}(\mathbb{R})$ sits in between: fine enough to encompass all intervals, open sets, and their countable combinations — coarse enough to remain consistent.

Imagine shading regions on a real line. The Borel σ-algebra is every region you can produce through interval operations and countable limits. Non-Borel sets require an uncountable number of individual choices to construct — they are artefacts of abstract set theory with no physical or financial interpretation.

A random variable maps states $\omega \in \Omega$ to numbers on the real line. Measurability ensures that for every interval $[a, b]$, the preimage — the collection of states mapping into $[a, b]$ — is one of the "allowed questions" in our σ-algebra.

Validation

The theoretical properties above can be checked computationally on finite sample spaces. The companion notebook verifies:

1. σ-algebra axioms — automated checks for complement closure and countable-union closure on $\Omega = \{\text{up}, \text{flat}, \text{down}\}$.
2. Generated σ-algebra — confirming $\sigma(X) = 2^\Omega$ when $X$ is injective, and $\sigma(X)$ has exactly $2^k$ elements when $X$ takes $k$ distinct values.
3. σ-additivity — verifying $\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B)$ for disjoint $A, B$ using exact rational arithmetic.
4. Borel generator equivalence — illustrating that $\sigma(\{(-\infty, x]\}) = \mathcal{B}(\mathbb{R})$ on a discretised real line.

Limitations

Non-measurable sets exist — they are just unpriceable. The Vitali set $V \subset [0,1]$ is constructed using the Axiom of Choice. If $V$ were Lebesgue-measurable with measure $\lambda(V)$: the unit interval decomposes as a countable union of translates of $V$, so $\lambda([0,1]) = \sum \lambda(V)$. If $\lambda(V) = 0$ the sum is $0 \ne 1$; if $\lambda(V) > 0$ the sum diverges. Contradiction in either case. Non-Borel sets play no role in practical finance but explain why "all subsets of $\mathbb{R}$" cannot serve as the event class.

The σ-algebra choice is a modelling commitment. Once you declare $(\Omega, \mathcal{F}, \mathbb{P})$, all subsequent pricing is relative to this structure. Enlarging $\mathcal{F}$ (e.g., with insider information) creates new admissible strategies and changes derivative prices — it is a fundamental modelling change, not a numerical parameter.

Finite spaces hide the real constraint. On a finite $\Omega$, the power set is always a valid σ-algebra and every function is measurable. The restriction to a proper sub-σ-algebra becomes binding only on uncountable spaces — in particular, on the path space of Brownian motion.

Path spaces require the Kolmogorov extension theorem. A Brownian motion path $\omega \mapsto (B_t(\omega))_{t \in [0,T]}$ lives in $C([0,T])$. The σ-algebra on this space is generated by cylinder sets $\{B_{t_1} \in A_1, \ldots, B_{t_n} \in A_n\}$. Its existence is guaranteed by the Kolmogorov extension theorem — but this is non-trivial and is the technical machinery underlying the construction of Brownian motion.