Probability Measures and Axioms

In sample-spaces-and-sigma-algebras we defined the sample space $\Omega$ and the $\sigma$-algebra $\mathcal{F}$ --- the structure of a probabilistic model. This article completes the triple $(\Omega ,\mathcal{F},P)$ by defining the probability measure $P$ itself.

A probability measure is a function $P:\mathcal{F}\to [0,1]$ that assigns a number to every event in a way that is internally consistent. Kolmogorov’s three axioms pin down exactly what “consistent” means, and from those axioms alone, a surprising amount of machinery follows: inclusion-exclusion, continuity of probability, Boole’s inequality, and the tools needed to construct measures on both finite and continuous spaces.

Without this article, the phrase “risk-neutral measure” is just a label. After it, you’ll understand precisely what it means for $Q$ to be a measure, what properties it must satisfy, and why the existence (or non-existence) of such a measure is equivalent to the absence of arbitrage.

Key Topics

• Kolmogorov’s three axioms: non-negativity, normalization, countable additivity
• Consequences of the axioms: inclusion-exclusion principle, continuity of probability (limits of increasing/decreasing sequences of events), Boole’s inequality
• Constructing measures on finite spaces: straightforward assignment of weights
• Constructing measures on continuous spaces: Lebesgue measure on $\mathbb{R}$, the Caratheodory extension theorem (how to go from a pre-measure on intervals to a full measure on Borel sets)
• Null sets and almost sure events: $P(A)=0$ does not mean $A=\varnothing$
• Absolute continuity and equivalence of measures: $Q\ll P$ means $Q$ agrees with $P$ on what’s possible --- the technical condition underlying Girsanov’s theorem

Finance Connections

• The Fundamental Theorem of Asset Pricing states: a market is arbitrage-free if and only if there exists a probability measure $Q$ equivalent to $P$ under which discounted prices are martingales. Understanding what “measure” and “equivalent” mean formally is prerequisite to understanding this theorem.
• Risk-neutral pricing requires computing $E^Q[\cdot ]$ --- an expectation under a specific measure. You need the axioms to know this is well-defined.

Prerequisites

• sample-spaces-and-sigma-algebras

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This article is a roadmap --- content to be developed in future sessions.