Probability Spaces

Every probability you ever compute lives on a probability space, a triple $(\Omega, \mathcal{F}, \mathbb{P})$. Most of the time you can ignore it, flip a coin, count outcomes, divide. But the moment you touch a continuum (a stock price that can be any real number, a Brownian path, a conditional expectation given an infinite past) the naive "favourable over total" definition collapses, and the machinery below is what keeps you from contradicting yourself. Understanding it is the difference between manipulating formulas and knowing why they are allowed.

The three ingredients

The sample space $\Omega$ is the set of all outcomes of the experiment. One toss of a coin: $\Omega = \{H, T\}$. A stock's terminal price: $\Omega = [0, \infty)$. A whole price path over $[0,T]$: $\Omega$ is a space of functions. An element $\omega \in \Omega$ is a single realisation of the entire experiment, a complete state of the world.

The σ-algebra $\mathcal{F}$ is the collection of subsets of $\Omega$ we are allowed to assign a probability to. Its members are called events. We require $\mathcal{F}$ to be a σ-algebra:

1. $\Omega \in \mathcal{F}$;
2. closed under complement: $A \in \mathcal{F} \Rightarrow A^c \in \mathcal{F}$;
3. closed under countable unions: $A_1, A_2, \dots \in \mathcal{F} \Rightarrow \bigcup_{n=1}^\infty A_n \in \mathcal{F}$.

De Morgan then gives closure under countable intersections for free. The word countable is load-bearing: it is exactly strong enough to handle limits ("the price eventually exceeds 100", "the average converges") without being so strong that no consistent measure exists.

The probability measure $\mathbb{P}$ is a function $\mathbb{P}: \mathcal{F} \to [0,1]$ satisfying Kolmogorov's axioms:

1. $\mathbb{P}(A) \ge 0$ for all $A \in \mathcal{F}$;
2. $\mathbb{P}(\Omega) = 1$;
3. countable additivity: for pairwise-disjoint $A_1, A_2, \dots$, $\mathbb{P}\!\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mathbb{P}(A_n).$

From these three lines everything follows: $\mathbb{P}(A^c) = 1 - \mathbb{P}(A)$, monotonicity ($A \subseteq B \Rightarrow \mathbb{P}(A) \le \mathbb{P}(B)$), inclusion–exclusion, and continuity from below/above ($A_n \uparrow A \Rightarrow \mathbb{P}(A_n) \to \mathbb{P}(A)$). That last property, continuity of measure, is the real payoff of demanding countable rather than merely finite additivity, and it is what lets you pass probability through limits.

Why measure theory at all?

Here is the failure that forces the whole apparatus. Suppose you want a "uniform" probability on $\Omega = [0,1]$ that is translation-invariant (rotating the circle shouldn't change lengths) and defined on every subset. No such measure exists. The Vitali construction uses the axiom of choice to build a set whose translated copies partition $[0,1]$ into countably many congruent pieces; countable additivity then forces its measure to be simultaneously $0$ and positive, a contradiction. The resolution is not to weaken the axioms but to shrink the domain: define $\mathbb{P}$ only on a σ-algebra of "nice" (measurable) sets, the Borel σ-algebra $\mathcal{B}([0,1])$ generated by the open intervals. Non-measurable sets exist; we simply never ask for their probability.

This is why you cannot get away with "probability = number of favourable outcomes / total". On a continuum every single outcome has probability zero ($\mathbb{P}(\{x\}) = 0$ for the uniform law), yet intervals have positive probability. Probability is fundamentally a property of sets of outcomes, assigned by a measure, not a headcount.

σ-algebras encode information

The subtlest role of $\mathcal{F}$ has nothing to do with pathological sets: a σ-algebra models what is knowable. A sub-σ-algebra $\mathcal{G} \subseteq \mathcal{F}$ represents the information available to an observer, the events they can decide "happened" or "didn't". Coarser $\mathcal{G}$ means less information. This is the entire foundation of Martingales and stochastic filtrations $\{\mathcal{F}_t\}$: $\mathcal{F}_t$ is "everything observable by time $t$", it grows with $t$ (you never forget), and a process is adapted if its value at $t$ is $\mathcal{F}_t$-measurable, i.e. knowable from the information you actually have. Conditional expectation $\mathbb{E}[X \mid \mathcal{G}]$, the cleanest object in probability, is defined as the best $\mathcal{G}$-measurable prediction of $X$. None of this parses without σ-algebras-as-information.

Worked example: the trivial vs. full information

Take one fair coin, $\Omega = \{H, T\}$. There are exactly two σ-algebras. The trivial one, $\mathcal{F}_0 = \{\varnothing, \Omega\}$, knows nothing: the only questions it can answer are "did something happen?" (yes, prob 1) and "did nothing happen?" (prob 0). The full one, $\mathcal{F}_1 = \{\varnothing, \{H\}, \{T\}, \Omega\}$, resolves the coin completely. A random variable $X(\omega)$ is measurable with respect to $\mathcal{F}_0$ only if it is constant, you cannot know the value of a non-constant function without information distinguishing $H$ from $T$. This toy case is the whole idea in miniature: measurability = knowability, and it scales directly to filtrations on price paths, where "$\mathcal{F}_t$-measurable" means "computable from the price history up to $t$".

Failure modes and subtleties

• Countable vs. finite additivity. Finitely-additive "probabilities" exist and behave strangely (they can fail continuity, allow a uniform distribution on the integers). Kolmogorov's countable additivity is a genuine, non-vacuous choice that rules these out and is what makes limit theorems work.
• Almost surely, not surely. Sets of measure zero are unavoidable and are routinely ignored: "$X = Y$ almost surely" means $\mathbb{P}(X \ne Y) = 0$, which is the strongest equality probability can offer on a continuum. Statements that hold only up to a null set are the norm, not a defect.
• The measure defines "impossible" loosely. $\mathbb{P}(A) = 0$ does not mean $A$ is empty (a continuous random variable hits any specific value with probability zero, yet hits some value). Conversely $\mathbb{P}(A)=1$ ("almost sure") does not mean $A = \Omega$.
• You rarely build $\mathcal{F}$ by hand. In practice you specify a generating class (intervals, cylinder sets) and invoke Carathéodory's extension theorem to get a unique measure on the σ-algebra they generate. Uniqueness needs a π-system argument (Dynkin's lemma), the reason two measures agreeing on intervals agree everywhere.

In interviews

You will rarely be asked to state Kolmogorov's axioms cold, but the space concept underpins the trick questions. A classic: "on a continuous distribution, what is $\mathbb{P}(X = 3)$?", zero, and the interviewer is checking you don't conflate density with probability. Another: "can every set have a probability?", no, and naming the Vitali/non-measurable-set reason signals real measure-theoretic literacy. If a stochastic-calculus or derivatives desk asks what a filtration is, the crisp answer is "an increasing family of σ-algebras modelling the information available over time, with respect to which the price process is adapted", connect it to Martingales and conditional expectation. The deepest signal you can send is knowing that probability is a measure on sets of outcomes, which is exactly why Random Variables & Distributions are defined as measurable functions on this space rather than as lists of values.