# Probability space
A probability space is the mathematical triple \((\Omega, \mathcal F, \mathbb P)\) on which probability theory is built: a sample space \(\Omega\) of possible outcomes, a σ-algebra \(\mathcal F\) of events whose probabilities can be assigned, and a probability measure \(\mathbb P\) that assigns each event a number between 0 and 1.[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) The formalization was introduced by [Andrey Kolmogorov](https://www.edgepedia.org/andrey-kolmogorov) in 1933 and places probability inside measure theory, the general theory of assigning sizes to sets.[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) • [2](https://openlibrary.org/books/OL13759409M/Foundations_of_the_theory_of_probability)
| Key fact | Detail |
|---|---|
| Definition | A triple \((\Omega, \mathcal F, \mathbb P)\): sample space, σ-algebra of events, and probability measure \(\mathbb P: \mathcal F \to [0,1]\)[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) • [3](https://archive.nptel.ac.in/content/storage2/courses/111101004/modules/lec1/1.6.html) |
| Measure axioms | \(\mathbb P(\Omega) = 1\), and \(\mathbb P\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} \mathbb P(A_n)\) for pairwise disjoint events[3](https://archive.nptel.ac.in/content/storage2/courses/111101004/modules/lec1/1.6.html) |
| σ-algebra closure | Contains \(\Omega\), closed under complements and countable unions[4](https://dictionary.helmholtz-uq.de/content/probab_space.html) |
| Extremes | Smallest σ-algebra is \(\{\emptyset, \Omega\}\); largest is the power set; for uncountable \(\Omega\) the power set can cause paradoxical results[4](https://dictionary.helmholtz-uq.de/content/probab_space.html) |
| Origin | Axiomatized by Kolmogorov in 1933 in *Foundations of the Theory of Probability*[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) • [2](https://openlibrary.org/books/OL13759409M/Foundations_of_the_theory_of_probability) |
| Key theorem | Kolmogorov's extension theorem: consistent finite-dimensional distributions determine a probability measure on an infinite product space[5](https://mathworld.wolfram.com/KolmogorovExtensionTheorem.html) |
| Random variable | A measurable function \(X: \Omega \to \mathbb R\) with \(\{X < a\} \in \mathcal F\) for every \(a\)[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) |
## The three components
**The sample space.** \(\Omega\) is a nonempty set whose elements are the outcomes of a random experiment; each element \(\omega \in \Omega\) is called an outcome.[3](https://archive.nptel.ac.in/content/storage2/courses/111101004/modules/lec1/1.6.html) • [6](https://www.statslab.cam.ac.uk/~mike/probability/notes.pdf) For a single coin toss \(\Omega = \{H, T\}\); for two independent tosses \(\Omega = \{HH, HT, TH, TT\}\); for a six-sided die \(\Omega = \{1, 2, \dots, 6\}\).[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) • [7](https://www.cs.cornell.edu/courses/cs2800/2017fa/lectures/lec04-probability.html)
**The σ-algebra of events.** \(\mathcal F\) is a collection of subsets of \(\Omega\), called events, on which probabilities are defined. It must be a σ-algebra (the terms σ-field and σ-algebra are interchangeable): \(\Omega \in \mathcal F\), the complement of any event in \(\mathcal F\) is again in \(\mathcal F\), and every countable union of events in \(\mathcal F\) belongs to \(\mathcal F\).[4](https://dictionary.helmholtz-uq.de/content/probab_space.html) • [6](https://www.statslab.cam.ac.uk/~mike/probability/notes.pdf) From closure under complements and countable unions it follows that countable intersections are also in \(\mathcal F\).[4](https://dictionary.helmholtz-uq.de/content/probab_space.html) The smallest possible σ-algebra is \(\{\emptyset, \Omega\}\), containing only the impossible event and the certain event; the largest is the power set \(\mathcal P(\Omega)\), the set of all subsets.[4](https://dictionary.helmholtz-uq.de/content/probab_space.html) • [1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html)
**The probability measure.** \(\mathbb P\) is a function \(\mathbb P: \mathcal F \to [0,1]\) satisfying two axioms: \(\mathbb P(\Omega) = 1\), and countable additivity, meaning that if \(A_1, A_2, \dots \in \mathcal F\) are pairwise disjoint then \(\mathbb P\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} \mathbb P(A_n)\).[3](https://archive.nptel.ac.in/content/storage2/courses/111101004/modules/lec1/1.6.html) In the equivalent formulation used in some courses, \(\mathbb P(\emptyset) = 0\), \(\mathbb P(\Omega) = 1\), and the countable-additivity property holds for any sequence of pairwise disjoint sets.[8](https://math.ucdavis.edu/~hunter/m280_09/ch.pdf) [Everything](https://www.edgepedia.org/everything) else known about probability, such as \(\mathbb P(\bar E) = 1 - \mathbb P(E)\) for a complement, is derived from these axioms.[7](https://www.cs.cornell.edu/courses/cs2800/2017fa/lectures/lec04-probability.html)
## Why not always use the power set
For a finite or countable sample space, taking \(\mathcal F = \mathcal P(\Omega)\) is perfectly adequate, and it is the natural choice in the standard examples: one fair coin toss uses \(\mathbb P(\emptyset) = 0\), \(\mathbb P(\{H\}) = \mathbb P(\{T\}) = 1/2\), \(\mathbb P(\Omega) = 1\), and two independent tosses use \(\mathbb P(A) = |A|/4\) for every \(A \subseteq \Omega\).[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html)
Uncountable sample spaces are different. To model drawing a real number uniformly from \([0,1]\), one wants \(\mathbb P([a,b]) = b - a\) for every \(a < b\) in \([0,1]\). A theorem recorded in the CMU notes states that no probability measure on \(([0,1], \mathcal P([0,1]))\) satisfies this; a smaller σ-algebra \(\mathcal L\) (the Lebesgue measurable sets) does support such a measure, which is the [Lebesgue measure](https://www.edgepedia.org/lebesgue-measure).[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) The same obstruction appears in the infinite coin-flip space: no probability measure on the full power set can assign each finite prefix of flips its probability \(1/2^N\), while a suitable σ-algebra does.[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) Assigning probabilities to all subsets of a continuous sample space can also produce highly paradoxical results, of which the [Banach–Tarski paradox](https://www.edgepedia.org/banach-tarski-paradox) is an example.[4](https://dictionary.helmholtz-uq.de/content/probab_space.html) A second reason for the event/outcome distinction is practical: for \(\Omega = [0,\ell]\) with all outcomes "equally probable", each single outcome has probability \(\mathbb P(\{x\}) = 0\), while intervals carry \(\mathbb P(a \le x \le b) = (b-a)/\ell\).[4](https://dictionary.helmholtz-uq.de/content/probab_space.html)
A related caution applies to finite spaces: \(\mathbb P(E) = |E|/|S|\) is true only for some probability spaces, namely those where all outcomes are equally likely, and assuming it generally leads to incorrect answers.[7](https://www.cs.cornell.edu/courses/cs2800/2017fa/lectures/lec04-probability.html)
## Relation to measure theory
This assimilation to measure theory is what allows the machinery of [Lebesgue integration](https://www.edgepedia.org/lebesgue-integration), product spaces, and convergence theorems to be applied to probability.[4](https://dictionary.helmholtz-uq.de/content/probab_space.html) • [10](https://arxiv.org/abs/2604.01685) In many applications the probability space is part of [Euclidean space](https://www.edgepedia.org/euclidean-space) and the σ-algebra is the smallest one containing all open sets, that is, the Borel σ-algebra.[11](https://people.math.harvard.edu/~knill/probability/index.html) When \(\mathcal F\) and \(\mathbb P\) are understood from context, authors often refer to the probability space simply as \(\Omega\).[8](https://math.ucdavis.edu/~hunter/m280_09/ch.pdf)
## Constructing infinite probability spaces
**Kolmogorov's extension theorem** answers the existence question for stochastic processes. In its general form, if a consistent family of finite-dimensional probability distributions is given on the finite products of the coordinate spaces, then there exists a probability measure \(\mathbb P\) on the product σ-algebra of the infinite product space that extends all of them; equivalently, there exists a probability space \((\Omega, \mathcal F, \mathbb P)\) carrying a stochastic process \(\{X_t, t \in T\}\) whose finite-dimensional distributions are the given ones.[5](https://mathworld.wolfram.com/KolmogorovExtensionTheorem.html) • [12](https://www.math.mcgill.ca/bill/589/589notes.pdf) A common version takes a sequence of probability measures \(\mu_1, \mu_2, \dots\) on the Borel sets of the real line and produces a probability space supporting an independent sequence of real random variables with those laws; this furnishes the technical foundation for discrete-time stochastic processes.[13](https://tropp.caltech.edu/notes/Tro24-Probability-Theory-LN.pdf) For example, choosing every \(\mu_i\) to be Bernoulli(1/2) yields a space supporting a countable sequence of independent fair coin flips.[13](https://tropp.caltech.edu/notes/Tro24-Probability-Theory-LN.pdf)
The construction has a structural limitation: in the product σ-algebra, every member depends on only countably many coordinates, so events such as "the sample path is continuous" are not events of the space at all and have no probability assigned.[12](https://www.math.mcgill.ca/bill/589/589notes.pdf) • [14](https://youyuanwu.github.io/misc-docs/math/11-probability/10-extension-theorem.html)
## Random variables on a probability space
A random variable on \((\Omega, \mathcal F, \mathbb P)\) is a function \(X: \Omega \to \mathbb R\) such that \(\{X < a\} \in \mathcal F\) for every real \(a\); that is, \(X\) is measurable with respect to \(\mathcal F\).[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) If \(\omega\) is the realized outcome, \(X(\omega)\) is called a realization of the random variable.[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) Sums, products, and continuous functions of random variables are again random variables.[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) The distribution of \(X\) is summarized by its cumulative distribution function \(F_X(x) = \mathbb P(X \le x)\).[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) In practice, once random variables and their laws are in hand, attention typically shifts away from the underlying space: the measure \(\mathbb P\) packs all the information about joint distributions, and the precise construction of \((\Omega, \mathcal F, \mathbb P)\) is rarely used again.[13](https://tropp.caltech.edu/notes/Tro24-Probability-Theory-LN.pdf)
## History
Kolmogorov's axiomatization appeared in his 1933 monograph *Foundations of the Theory of Probability*, written in German and later translated into English, with a second English edition published by Chelsea Publishing Company in 1956.[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) • [2](https://openlibrary.org/books/OL13759409M/Foundations_of_the_theory_of_probability) • [15](https://cml.rhul.ac.uk/resources/fop/index.htm) Before this level of rigor, informal probability arguments could produce nonsense, as in the [two envelopes problem](https://www.edgepedia.org/two-envelopes-problem), where plausible reasoning about an unknown amount of money suggests both that switching is pointless and that switching always helps.[1](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html)
## References
1. [Probability spaces and Random Variables, CMU 21-425 lecture notes](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html)
2. [Foundations of the theory of probability, A. N. Kolmogorov, Open Library record](https://openlibrary.org/books/OL13759409M/Foundations_of_the_theory_of_probability)
3. [NPTEL Probability and Stochastic Processes, Definitions 1.6–1.7](https://archive.nptel.ac.in/content/storage2/courses/111101004/modules/lec1/1.6.html)
4. [Probability Spaces and Random Variables, Helmholtz Uncertainty Quantification dictionary](https://dictionary.helmholtz-uq.de/content/probab_space.html)
5. [Kolmogorov Extension Theorem, Wolfram MathWorld](https://mathworld.wolfram.com/KolmogorovExtensionTheorem.html)
6. [Introduction to Probability, Cambridge Stats Lab lecture notes (M. Tehranchi, Michaelmas 2006)](https://www.statslab.cam.ac.uk/~mike/probability/notes.pdf)
7. [Lecture 4: Probability, CS 2800, Cornell University](https://www.cs.cornell.edu/courses/cs2800/2017fa/lectures/lec04-probability.html)
8. [Lecture Notes on Measure and Probability, UC Davis (T. Hunter)](https://math.ucdavis.edu/~hunter/m280_09/ch.pdf)
9. [Advanced Probability lecture notes, University of Vienna](https://www.mat.univie.ac.at/~mathias/Advanced_Probability.pdf)
10. [Lecture notes: Probability with Measure, arXiv](https://arxiv.org/abs/2604.01685)
11. [Probability Theory and Stochastic Processes with Applications, Harvard (P. Knill)](https://people.math.harvard.edu/~knill/probability/index.html)
12. [Chapter 6: The Kolmogorov Extension Theorem, McGill 589 notes](https://www.math.mcgill.ca/bill/589/589notes.pdf)
13. [Probability Theory & Stochastic Processes, Caltech CMS 117 notes (J. Tropp)](https://tropp.caltech.edu/notes/Tro24-Probability-Theory-LN.pdf)
14. [The Kolmogorov Extension Theorem, notes by Youyuan Wu](https://youyuanwu.github.io/misc-docs/math/11-probability/10-extension-theorem.html)
15. [A. N. Kolmogorov's Foundations of the Theory of Probability, CLRC download page](https://cml.rhul.ac.uk/resources/fop/index.htm)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Probability spaces and axioms*
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